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Theorem ax12 2212
Description: Rederivation of axiom ax-12 1794 from ax-c15 2198, ax-c11 2196, and other older axioms. See theorem axc15 2042 for the derivation of ax-c15 2198 from ax-12 1794.

An open problem is whether we can prove this using ax-c11n 2197 instead of ax-c11 2196.

This proof uses newer axioms ax-4 1603 and ax-6 1710, 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 2193 and ax-c10 2195. (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 237 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ph ) )
21dral1-o 2211 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ph ) )
3 ax-1 6 . . . . 5  |-  ( ph  ->  ( x  =  y  ->  ph ) )
43alimi 1605 . . . 4  |-  ( A. x ph  ->  A. x
( x  =  y  ->  ph ) )
52, 4syl6bir 229 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
) )
65a1d 25 . 2  |-  ( A. x  x  =  y  ->  ( x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
) ) )
7 ax-c5 2192 . . 3  |-  ( A. y ph  ->  ph )
8 ax-c15 2198 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
97, 8syl7 68 . 2  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
106, 9pm2.61i 164 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 1368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-11 1782  ax-c5 2192  ax-c4 2193  ax-c7 2194  ax-c11 2196  ax-c15 2198  ax-c9 2199
This theorem depends on definitions:  df-bi 185  df-ex 1588
This theorem is referenced by:  axc11-o  2259
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