# Metamath Proof Explorer

## Theorem ax12a2-o

Description: Derive ax-c15 from a hypothesis in the form of ax-12 , without using ax-12 or ax-c15 . The hypothesis is weaker than ax-12 , with z both distinct from x and not occurring in ph . Thus, the hypothesis provides an alternate axiom that can be used in place of ax-12 , if we also have ax-c11 , which this proof uses. As theorem ax12 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n instead of ax-c11 . (Contributed by NM, 2-Feb-2007) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypothesis ax12a2-o.1 ${⊢}{x}={z}\to \left(\forall {z}\phantom{\rule{.4em}{0ex}}{\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={z}\to {\phi }\right)\right)$
Assertion ax12a2-o ${⊢}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \left({x}={y}\to \left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)\right)\right)$

### Proof

Step Hyp Ref Expression
1 ax12a2-o.1 ${⊢}{x}={z}\to \left(\forall {z}\phantom{\rule{.4em}{0ex}}{\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={z}\to {\phi }\right)\right)$
2 ax-5 ${⊢}{\phi }\to \forall {z}\phantom{\rule{.4em}{0ex}}{\phi }$
3 2 1 syl5 ${⊢}{x}={z}\to \left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={z}\to {\phi }\right)\right)$
4 3 ax12v2-o ${⊢}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \left({x}={y}\to \left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)\right)\right)$