Metamath Proof Explorer

Theorem cbvmpt

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011) Add disjoint variable condition to avoid ax-13 . See cbvmptg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024)

Ref Expression
Hypotheses cbvmpt.1 _ y B
cbvmpt.2 _ x C
cbvmpt.3 x = y B = C
Assertion cbvmpt x A B = y A C


Step Hyp Ref Expression
1 cbvmpt.1 _ y B
2 cbvmpt.2 _ x C
3 cbvmpt.3 x = y B = C
4 nfcv _ x A
5 nfcv _ y A
6 4 5 1 2 3 cbvmptf x A B = y A C