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Theorem axc15 2141
Description: Derivation of set.mm's original ax-c15 32386 from ax-c11n 32385 and the shorter ax-12 1906 that has replaced it.

Theorem ax12 32400 shows the reverse derivation of ax-12 1906 from ax-c15 32386.

Normally, axc15 2141 should be used rather than ax-c15 32386, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)

Assertion
Ref Expression
axc15  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )

Proof of Theorem axc15
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ax-12 1906 . 2  |-  ( x  =  z  ->  ( A. z ph  ->  A. x
( x  =  z  ->  ph ) ) )
21ax12a2 2140 1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  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-10 1888  ax-12 1906  ax-13 2054
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1661  df-nf 1665
This theorem is referenced by:  ax12b  2142  equs5  2146  ax12vALT  2223
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