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Theorem ax11v2 1748
Description: Recovery of ax11o 1750 from ax11v 1755 without using ax-11 1442. The hypothesis is even weaker than ax11v 1755, 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 ax11o 1750. (Contributed by NM, 2-Feb-2007.)
Hypothesis
Ref Expression
ax11v2.1  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
Assertion
Ref Expression
ax11v2  |-  ( -. 
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
StepHypRef Expression
1 a9e 1631 . 2  |-  E. z 
z  =  y
2 ax11v2.1 . . . . 5  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
3 equequ2 1646 . . . . . . 7  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
43adantl 271 . . . . . 6  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  -> 
( x  =  z  <-> 
x  =  y ) )
5 dveeq2 1743 . . . . . . . . 9  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
65imp 122 . . . . . . . 8  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  ->  A. x  z  =  y )
7 hba1 1478 . . . . . . . . 9  |-  ( A. x  z  =  y  ->  A. x A. x  z  =  y )
83imbi1d 229 . . . . . . . . . 10  |-  ( z  =  y  ->  (
( x  =  z  ->  ph )  <->  ( x  =  y  ->  ph )
) )
98sps 1475 . . . . . . . . 9  |-  ( A. x  z  =  y  ->  ( ( x  =  z  ->  ph )  <->  ( x  =  y  ->  ph )
) )
107, 9albidh 1414 . . . . . . . 8  |-  ( A. x  z  =  y  ->  ( A. x ( x  =  z  ->  ph )  <->  A. x ( x  =  y  ->  ph )
) )
116, 10syl 14 . . . . . . 7  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  -> 
( A. x ( x  =  z  ->  ph )  <->  A. x ( x  =  y  ->  ph )
) )
1211imbi2d 228 . . . . . 6  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  -> 
( ( ph  ->  A. x ( x  =  z  ->  ph ) )  <-> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
134, 12imbi12d 232 . . . . 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 146 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  z  =  y )  -> 
( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
1514ex 113 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) ) )
1615exlimdv 1747 . 2  |-  ( -. 
A. x  x  =  y  ->  ( E. z  z  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) ) )
171, 16mpi 15 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    /\ wa 102    <-> wb 103   A.wal 1287   E.wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by:  ax11a2  1749
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