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

Proof

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