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Theorem ax12 32400
Description: Rederivation of axiom ax-12 1906 from ax-c15 32386, ax-c11 32384, and other older axioms. See theorem axc15 2141 for the derivation of ax-c15 32386 from ax-12 1906.

An open problem is whether we can prove this using ax-c11n 32385 instead of ax-c11 32384.

This proof uses newer axioms ax-4 1679 and ax-6 1795, 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 32381 and ax-c10 32383. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
ax12  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )

Proof of Theorem ax12
StepHypRef Expression
1 biidd 241 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ph ) )
21dral1-o 32399 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ph ) )
3 ax-1 6 . . . . 5  |-  ( ph  ->  ( x  =  y  ->  ph ) )
43alimi 1681 . . . 4  |-  ( A. x ph  ->  A. x
( x  =  y  ->  ph ) )
52, 4syl6bir 233 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
) )
65a1d 27 . 2  |-  ( A. x  x  =  y  ->  ( x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
) ) )
7 ax-c5 32380 . . 3  |-  ( A. y ph  ->  ph )
8 ax-c15 32386 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
97, 8syl7 71 . 2  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
106, 9pm2.61i 168 1  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-11 1893  ax-c5 32380  ax-c4 32381  ax-c7 32382  ax-c11 32384  ax-c15 32386  ax-c9 32387
This theorem depends on definitions:  df-bi 189  df-ex 1661
This theorem is referenced by:  axc11-o  32447
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