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Metamath Proof Explorer

Theorem cbvmptvg

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See cbvmptv for a version with more disjoint variable conditions, but not requiring ax-13 . (Contributed by Mario Carneiro, 19-Feb-2013) (New usage is discouraged.)

Ref Expression
Hypothesis cbvmptvg.1 
|- ( x = y -> B = C )
Assertion cbvmptvg 
|- ( x e. A |-> B ) = ( y e. A |-> C )

Proof

Step Hyp Ref Expression
1 cbvmptvg.1 
 |-  ( x = y -> B = C )
2 nfcv 
 |-  F/_ y B
3 nfcv 
 |-  F/_ x C
4 2 3 1 cbvmptg 
 |-  ( x e. A |-> B ) = ( y e. A |-> C )