# Metamath Proof Explorer

## Theorem nfcvf

Description: If x and y are distinct, then x is not free in y . Usage of this theorem is discouraged because it depends on ax-13 . See nfcv for a version that replaces the distinctor with a disjoint variable condition, requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016) Avoid ax-ext . (Revised by Wolf Lammen, 10-May-2023) (New usage is discouraged.)

Ref Expression
Assertion nfcvf ${⊢}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{y}$

### Proof

Step Hyp Ref Expression
1 nfv ${⊢}Ⅎ{w}\phantom{\rule{.4em}{0ex}}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}$
2 nfv ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{w}\in {z}$
3 elequ2 ${⊢}{z}={y}\to \left({w}\in {z}↔{w}\in {y}\right)$
4 2 3 dvelimnf ${⊢}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to Ⅎ{x}\phantom{\rule{.4em}{0ex}}{w}\in {y}$
5 1 4 nfcd ${⊢}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{y}$