# Metamath Proof Explorer

## Theorem sb3b

Description: Simplified definition of substitution when variables are distinct. This is the biconditional strengthening of sb3 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by BJ, 6-Oct-2018) Shorten sb3 . (Revised by Wolf Lammen, 21-Feb-2021) (New usage is discouraged.)

Ref Expression
Assertion sb3b ${⊢}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \left(\left[{y}/{x}\right]{\phi }↔\exists {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\wedge {\phi }\right)\right)$

### Proof

Step Hyp Ref Expression
1 sb4b ${⊢}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \left(\left[{y}/{x}\right]{\phi }↔\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)\right)$
2 equs5 ${⊢}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \left(\exists {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\wedge {\phi }\right)↔\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)\right)$
3 1 2 bitr4d ${⊢}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \left(\left[{y}/{x}\right]{\phi }↔\exists {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\wedge {\phi }\right)\right)$