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Theorem ax12v2-o 32232
Description: Recovery of ax-c15 32173 from ax12v 1908 without using ax-c15 32173. The hypothesis is even weaker than ax12v 1908, with  z both distinct from  x and not occurring in  ph. Thus, the hypothesis provides an alternate axiom that can be used in place of ax-c15 32173. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax12v2-o.1  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
Assertion
Ref Expression
ax12v2-o  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
Distinct variable groups:    x, z    y, z    ph, z
Allowed substitution hints:    ph( x, y)

Proof of Theorem ax12v2-o
StepHypRef Expression
1 ax6ev 1799 . 2  |-  E. z 
z  =  y
2 ax12v2-o.1 . . . . 5  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
3 equequ2 1851 . . . . . . 7  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
43adantl 467 . . . . . 6  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  -> 
( x  =  z  <-> 
x  =  y ) )
5 dveeq2-o 32216 . . . . . . . . 9  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
65imp 430 . . . . . . . 8  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  ->  A. x  z  =  y )
7 nfa1-o 32198 . . . . . . . . 9  |-  F/ x A. x  z  =  y
83imbi1d 318 . . . . . . . . . 10  |-  ( z  =  y  ->  (
( x  =  z  ->  ph )  <->  ( x  =  y  ->  ph )
) )
98sps-o 32191 . . . . . . . . 9  |-  ( A. x  z  =  y  ->  ( ( x  =  z  ->  ph )  <->  ( x  =  y  ->  ph )
) )
107, 9albid 1938 . . . . . . . 8  |-  ( A. x  z  =  y  ->  ( A. x ( x  =  z  ->  ph )  <->  A. x ( x  =  y  ->  ph )
) )
116, 10syl 17 . . . . . . 7  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  -> 
( A. x ( x  =  z  ->  ph )  <->  A. x ( x  =  y  ->  ph )
) )
1211imbi2d 317 . . . . . 6  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  -> 
( ( ph  ->  A. x ( x  =  z  ->  ph ) )  <-> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
134, 12imbi12d 321 . . . . 5  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  -> 
( ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )  <->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) ) )
142, 13mpbii 214 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  -> 
( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
1514ex 435 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) ) )
1615exlimdv 1771 . 2  |-  ( -. 
A. x  x  =  y  ->  ( E. z  z  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) ) )
171, 16mpi 21 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    <-> wb 187    /\ wa 370   A.wal 1435   E.wex 1659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-c5 32167  ax-c4 32168  ax-c7 32169  ax-c11 32171  ax-c9 32174
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664
This theorem is referenced by:  ax12a2-o  32233
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