Apparently Mathematica provides syntax for extracting conditional CDFs from a continuous joint distribution:

This syntax can be used as an intermediate stepping stone to deriving the desired conditional PDF.

dist = ProbabilityDistribution[Piecewise[{{1/\[Pi], 0 <= Sqrt[x^2 + y^2] <= 1}}], {x, -1,1}, {y, -1, 1}];
cdf = Probability[X <= x \[Conditioned] Y == y, {X, Y} \[Distributed] dist];
conditionalDensityFunction = PDF[ProbabilityDistribution[{"CDF", cdf}, {x, -1, 1}], x] 

#update

The comments underneath this answer to a similar question suggests that we are still waiting for this functionality:

Apparently Mathematica provides syntax for extracting conditional CDFs from a continuous joint distribution:

This syntax can be used as an intermediate stepping stone to deriving the desired conditional PDF.

dist = ProbabilityDistribution[Piecewise[{{1/\[Pi], 0 <= Sqrt[x^2 + y^2] <= 1}}], {x, -1,1}, {y, -1, 1}];
cdf = Probability[X <= x \[Conditioned] Y == y, {X, Y} \[Distributed] dist];
conditionalDensityFunction = PDF[ProbabilityDistribution[{"CDF", cdf}, {x, -1, 1}], x] 

update

The comments underneath this answer to a similar question suggests that we are still waiting for this functionality:

Apparently Mathematica provides syntax for extracting conditional CDFs from a continuous joint distribution:

This syntax can be used as an intermediate stepping stone to deriving the desired conditional PDF.

dist = ProbabilityDistribution[Piecewise[{{1/\[Pi], 0 <= Sqrt[x^2 + y^2] <= 1}}], {x, -1,1}, {y, -1, 1}];
cdf = Probability[X <= x \[Conditioned] Y == y, {X, Y} \[Distributed] dist];
conditionalDensityFunction = PDF[ProbabilityDistribution[{"CDF", cdf}, {x, -1, 1}], x] 

#update

The comments underneath this answer to a similar question suggests that we are still waiting for this functionality:

Apparently Mathematica provides syntax for extracting conditional CDFs from a continuous joint distribution:

This syntax can be used as an intermediate stepping stone to deriving the desired conditional PDF.

dist = ProbabilityDistribution[Piecewise[{{1/\[Pi], 0 <= Sqrt[x^2 + y^2] <= 1}}], {x, -1,1}, {y, -1, 1}];
cdf = Probability[X <= x \[Conditioned] Y == y, {X, Y} \[Distributed] dist];
conditionalDensityFunction = PDF[ProbabilityDistribution[{"CDF", cdf}, {x, -1, 1}], x] 

#update

The comments underneath this answer to a similar question suggests that we are still waiting for this functionality:

Apparently Mathematica provides syntax for extracting conditional CDFs from a continuous joint distribution:

This syntax can be used as an intermediate stepping stone to deriving the desired conditional PDF.

dist = ProbabilityDistribution[Piecewise[{{1/\[Pi], 0 <= Sqrt[x^2 + y^2] <= 1}}], {x, -1,1}, {y, -1, 1}];
cdf = Probability[X <= x \[Conditioned] Y == y, {X, Y} \[Distributed] dist];
conditionalDensityFunction =PDF[ProbabilityDistribution[{"CDF", cdf}, {x, -1, 1}, {y, -1, 1}] , {x, y}]

#update

The comments underneath this answer to a similar question suggests that we are still waiting for this functionality:

Apparently Mathematica provides syntax for extracting conditional CDFs from a continuous joint distribution:

This syntax can be used as an intermediate stepping stone to deriving the desired conditional PDF.

dist = ProbabilityDistribution[Piecewise[{{1/\[Pi], 0 <= Sqrt[x^2 + y^2] <= 1}}], {x, -1,1}, {y, -1, 1}];
cdf = Probability[X <= x \[Conditioned] Y == y, {X, Y} \[Distributed] dist];
conditionalDensityFunction = PDF[ProbabilityDistribution[{"CDF", cdf}, {x, -1, 1}], x] 

#update

The comments underneath this answer to a similar question suggests that we are still waiting for this functionality: