Metamath Proof Explorer


Theorem axc11-o

Description: Show that ax-c11 can be derived from ax-c11n and ax-12 . An open problem is whether this theorem can be derived from ax-c11n and the others when ax-12 is replaced with ax-c15 or ax12v . See Theorems axc11nfromc11 for the rederivation of ax-c11n from axc11 .

Normally, axc11 should be used rather than ax-c11 or axc11-o , except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion axc11-o
|- ( A. x x = y -> ( A. x ph -> A. y ph ) )

Proof

Step Hyp Ref Expression
1 ax-c11n
 |-  ( A. x x = y -> A. y y = x )
2 ax12
 |-  ( y = x -> ( A. x ph -> A. y ( y = x -> ph ) ) )
3 2 equcoms
 |-  ( x = y -> ( A. x ph -> A. y ( y = x -> ph ) ) )
4 3 sps-o
 |-  ( A. x x = y -> ( A. x ph -> A. y ( y = x -> ph ) ) )
5 pm2.27
 |-  ( y = x -> ( ( y = x -> ph ) -> ph ) )
6 5 al2imi
 |-  ( A. y y = x -> ( A. y ( y = x -> ph ) -> A. y ph ) )
7 1 4 6 sylsyld
 |-  ( A. x x = y -> ( A. x ph -> A. y ph ) )