Theorem cbvmptv

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013) Add disjoint variable condition to avoid ax-13 . See cbvmptvg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024)

	Ref	Expression
Hypothesis	cbvmptv.1	$$⊢x=y\to B=C$$
Assertion	cbvmptv	$$⊢\left(x\in A⟼B\right)=\left(y\in A⟼C\right)$$

Proof

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