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Theorem ax9lem3 29142
Description: Lemma for ax9 1889. Similar to sp 1716, without using sp 1716, ax9 1889, or ax10 1884. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ax9lem3.a  |-  -.  A. w  -.  w  =  x
ax9lem3.c  |-  -.  A. x  -.  x  =  w
Assertion
Ref Expression
ax9lem3  |-  ( A. x ph  ->  ph )
Distinct variable groups:    x, w    ph, w
Allowed substitution hint:    ph( x)

Proof of Theorem ax9lem3
StepHypRef Expression
1 ax9lem3.a . 2  |-  -.  A. w  -.  w  =  x
2 ax-17 1603 . . 3  |-  ( -.  ( A. x ph  ->  ph )  ->  A. w  -.  ( A. x ph  ->  ph ) )
3 ax9lem3.c . . . . . . . 8  |-  -.  A. x  -.  x  =  w
43ax9lem1 29140 . . . . . . 7  |-  ( w  =  x  ->  x  =  w )
5 ax-17 1603 . . . . . . 7  |-  ( -. 
ph  ->  A. w  -.  ph )
6 ax-11 1715 . . . . . . 7  |-  ( x  =  w  ->  ( A. w  -.  ph  ->  A. x ( x  =  w  ->  -.  ph )
) )
74, 5, 6syl2im 34 . . . . . 6  |-  ( w  =  x  ->  ( -.  ph  ->  A. x
( x  =  w  ->  -.  ph ) ) )
8 con2 108 . . . . . . . 8  |-  ( ( x  =  w  ->  -.  ph )  ->  ( ph  ->  -.  x  =  w ) )
98al2imi 1548 . . . . . . 7  |-  ( A. x ( x  =  w  ->  -.  ph )  ->  ( A. x ph  ->  A. x  -.  x  =  w ) )
103, 9mtoi 169 . . . . . 6  |-  ( A. x ( x  =  w  ->  -.  ph )  ->  -.  A. x ph )
117, 10syl6 29 . . . . 5  |-  ( w  =  x  ->  ( -.  ph  ->  -.  A. x ph ) )
1211con4d 97 . . . 4  |-  ( w  =  x  ->  ( A. x ph  ->  ph )
)
1312con3i 127 . . 3  |-  ( -.  ( A. x ph  ->  ph )  ->  -.  w  =  x )
142, 13alrimih 1552 . 2  |-  ( -.  ( A. x ph  ->  ph )  ->  A. w  -.  w  =  x
)
151, 14mt3 171 1  |-  ( A. x ph  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527
This theorem is referenced by:  ax9lem4  29143  ax9lem7  29146  ax9lem8  29147  ax9lem9  29148  ax9lem12  29151  ax9lem13  29152  ax9lem15  29154  ax9lem17  29156  ax9vax9  29158
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-8 1643  ax-11 1715
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