Metamath Proof Explorer


Theorem ax12fromc15

Description: Rederivation of Axiom ax-12 from ax-c15 , ax-c11 (used through dral1-o ), and other older axioms. See Theorem axc15 for the derivation of ax-c15 from ax-12 .

An open problem is whether we can prove this using ax-c11n instead of ax-c11 .

This proof uses newer axioms ax-4 and ax-6 , but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 and ax-c10 . (Contributed by NM, 22-Jan-2007) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion ax12fromc15
|- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )

Proof

Step Hyp Ref Expression
1 biidd
 |-  ( A. x x = y -> ( ph <-> ph ) )
2 1 dral1-o
 |-  ( A. x x = y -> ( A. x ph <-> A. y ph ) )
3 ax-1
 |-  ( ph -> ( x = y -> ph ) )
4 3 alimi
 |-  ( A. x ph -> A. x ( x = y -> ph ) )
5 2 4 syl6bir
 |-  ( A. x x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
6 5 a1d
 |-  ( A. x x = y -> ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) )
7 ax-c5
 |-  ( A. y ph -> ph )
8 ax-c15
 |-  ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
9 7 8 syl7
 |-  ( -. A. x x = y -> ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) )
10 6 9 pm2.61i
 |-  ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )