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Theorem ax11v2-o 2140
Description: Recovery of ax-11o 2080 from ax11v 2036 without using ax-11o 2080. The hypothesis is even weaker than ax11v 2036, 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-11o 2080. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax11v2-o.1  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
Assertion
Ref Expression
ax11v2-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 ax11v2-o
StepHypRef Expression
1 a9ev 1637 . 2  |-  E. z 
z  =  y
2 ax11v2-o.1 . . . . 5  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
3 equequ2 1649 . . . . . . 7  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
43adantl 452 . . . . . 6  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  -> 
( x  =  z  <-> 
x  =  y ) )
5 dveeq2-o 2123 . . . . . . . . 9  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
65imp 418 . . . . . . . 8  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  ->  A. x  z  =  y )
7 nfa1-o 2105 . . . . . . . . 9  |-  F/ x A. x  z  =  y
83imbi1d 308 . . . . . . . . . 10  |-  ( z  =  y  ->  (
( x  =  z  ->  ph )  <->  ( x  =  y  ->  ph )
) )
98sps-o 2098 . . . . . . . . 9  |-  ( A. x  z  =  y  ->  ( ( x  =  z  ->  ph )  <->  ( x  =  y  ->  ph )
) )
107, 9albid 1752 . . . . . . . 8  |-  ( A. x  z  =  y  ->  ( A. x ( x  =  z  ->  ph )  <->  A. x ( x  =  y  ->  ph )
) )
116, 10syl 15 . . . . . . 7  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  -> 
( A. x ( x  =  z  ->  ph )  <->  A. x ( x  =  y  ->  ph )
) )
1211imbi2d 307 . . . . . 6  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  -> 
( ( ph  ->  A. x ( x  =  z  ->  ph ) )  <-> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
134, 12imbi12d 311 . . . . 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 202 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  -> 
( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
1514ex 423 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) ) )
1615exlimdv 1664 . 2  |-  ( -. 
A. x  x  =  y  ->  ( E. z  z  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) ) )
171, 16mpi 16 1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528
This theorem is referenced by:  ax11a2-o  2141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-4 2074  ax-5o 2075  ax-6o 2076  ax-10o 2078  ax-12o 2081
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
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