# Metamath Proof Explorer

## Theorem cbvmptg

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. Usage of this theorem is discouraged because it depends on ax-13 . See cbvmpt for a version with more disjoint variable conditions, but not requiring ax-13 . (Contributed by NM, 11-Sep-2011) (New usage is discouraged.)

Ref Expression
Hypotheses cbvmptg.1 ${⊢}\underset{_}{Ⅎ}{y}\phantom{\rule{.4em}{0ex}}{B}$
cbvmptg.2 ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{C}$
cbvmptg.3 ${⊢}{x}={y}\to {B}={C}$
Assertion cbvmptg ${⊢}\left({x}\in {A}⟼{B}\right)=\left({y}\in {A}⟼{C}\right)$

### Proof

Step Hyp Ref Expression
1 cbvmptg.1 ${⊢}\underset{_}{Ⅎ}{y}\phantom{\rule{.4em}{0ex}}{B}$
2 cbvmptg.2 ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{C}$
3 cbvmptg.3 ${⊢}{x}={y}\to {B}={C}$
4 nfcv ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{A}$
5 nfcv ${⊢}\underset{_}{Ⅎ}{y}\phantom{\rule{.4em}{0ex}}{A}$
6 4 5 1 2 3 cbvmptfg ${⊢}\left({x}\in {A}⟼{B}\right)=\left({y}\in {A}⟼{C}\right)$