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Theorem ax11v2 2048
Description: Recovery of ax-11o 2195 from ax11v 2149. This proof uses ax-10 2194 and ax-11 1757. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)
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 a9ev 1664 . 2  |-  E. z 
z  =  y
2 dveeq2 2022 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
3 ax11v2.1 . . . . 5  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
4 equequ2 1694 . . . . . . 7  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
54sps 1766 . . . . . 6  |-  ( A. x  z  =  y  ->  ( x  =  z  <-> 
x  =  y ) )
6 nfa1 1802 . . . . . . . 8  |-  F/ x A. x  z  =  y
75imbi1d 309 . . . . . . . 8  |-  ( A. x  z  =  y  ->  ( ( x  =  z  ->  ph )  <->  ( x  =  y  ->  ph )
) )
86, 7albid 1784 . . . . . . 7  |-  ( A. x  z  =  y  ->  ( A. x ( x  =  z  ->  ph )  <->  A. x ( x  =  y  ->  ph )
) )
98imbi2d 308 . . . . . 6  |-  ( A. x  z  =  y  ->  ( ( ph  ->  A. x ( x  =  z  ->  ph ) )  <-> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
105, 9imbi12d 312 . . . . 5  |-  ( A. x  z  =  y  ->  ( ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )  <->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) ) )
113, 10mpbii 203 . . . 4  |-  ( A. x  z  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
122, 11syl6 31 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) ) )
1312exlimdv 1643 . 2  |-  ( -. 
A. x  x  =  y  ->  ( E. z  z  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) ) )
141, 13mpi 17 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 177   A.wal 1546   E.wex 1547
This theorem is referenced by:  ax11a2  2050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551
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