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	$$⊢x=y\to B=C$$
Assertion	cbvmptvg	$$⊢\left(x\in A⟼B\right)=\left(y\in A⟼C\right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvmptvg.1	$$⊢x=y\to B=C$$
2		nfcv	$$⊢\underset{\_}{Ⅎ}yB$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}xC$$
4	2 3 1	cbvmptg	$$⊢\left(x\in A⟼B\right)=\left(y\in A⟼C\right)$$