Metamath Proof Explorer

Theorem ax13b

Description: An equivalence between two ways of expressing ax-13 . See the comment for ax-13 . (Contributed by NM, 2-May-2017) (Proof shortened by Wolf Lammen, 26-Feb-2018) (Revised by BJ, 15-Sep-2020)

Ref Expression
Assertion ax13b ${⊢}\left(¬{x}={y}\to \left({y}={z}\to {\phi }\right)\right)↔\left(¬{x}={y}\to \left(¬{x}={z}\to \left({y}={z}\to {\phi }\right)\right)\right)$

Proof

Step Hyp Ref Expression
1 ax-1 ${⊢}\left({y}={z}\to {\phi }\right)\to \left(¬{x}={z}\to \left({y}={z}\to {\phi }\right)\right)$
2 equeuclr ${⊢}{y}={z}\to \left({x}={z}\to {x}={y}\right)$
3 2 con3rr3 ${⊢}¬{x}={y}\to \left({y}={z}\to ¬{x}={z}\right)$
4 3 imim1d ${⊢}¬{x}={y}\to \left(\left(¬{x}={z}\to \left({y}={z}\to {\phi }\right)\right)\to \left({y}={z}\to \left({y}={z}\to {\phi }\right)\right)\right)$
5 pm2.43 ${⊢}\left({y}={z}\to \left({y}={z}\to {\phi }\right)\right)\to \left({y}={z}\to {\phi }\right)$
6 4 5 syl6 ${⊢}¬{x}={y}\to \left(\left(¬{x}={z}\to \left({y}={z}\to {\phi }\right)\right)\to \left({y}={z}\to {\phi }\right)\right)$
7 1 6 impbid2 ${⊢}¬{x}={y}\to \left(\left({y}={z}\to {\phi }\right)↔\left(¬{x}={z}\to \left({y}={z}\to {\phi }\right)\right)\right)$
8 7 pm5.74i ${⊢}\left(¬{x}={y}\to \left({y}={z}\to {\phi }\right)\right)↔\left(¬{x}={y}\to \left(¬{x}={z}\to \left({y}={z}\to {\phi }\right)\right)\right)$