Metamath Proof Explorer

Theorem ax12b

Description: A bidirectional version of axc15 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 30-Jun-2006) (New usage is discouraged.)

Ref Expression
Assertion ax12b ${⊢}\left(¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\wedge {x}={y}\right)\to \left({\phi }↔\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)\right)$

Proof

Step Hyp Ref Expression
1 axc15 ${⊢}¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \left({x}={y}\to \left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)\right)\right)$
2 1 imp ${⊢}\left(¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\wedge {x}={y}\right)\to \left({\phi }\to \forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)\right)$
3 sp ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)\to \left({x}={y}\to {\phi }\right)$
4 3 com12 ${⊢}{x}={y}\to \left(\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)\to {\phi }\right)$
5 4 adantl ${⊢}\left(¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\wedge {x}={y}\right)\to \left(\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)\to {\phi }\right)$
6 2 5 impbid ${⊢}\left(¬\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\wedge {x}={y}\right)\to \left({\phi }↔\forall {x}\phantom{\rule{.4em}{0ex}}\left({x}={y}\to {\phi }\right)\right)$