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Theorem ax9lem4 29143
Description: Lemma for ax9 1889. Similar to ax9o 1890, 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
ax9lem4.a  |-  -.  A. w  -.  w  =  x
ax9lem4.c  |-  -.  A. x  -.  x  =  w
ax9lem4.d  |-  -.  A. x  -.  x  =  y
Assertion
Ref Expression
ax9lem4  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )
Distinct variable groups:    x, w    ph, w    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem ax9lem4
StepHypRef Expression
1 ax9lem4.d . . 3  |-  -.  A. x  -.  x  =  y
2 con3 126 . . . 4  |-  ( ( x  =  y  ->  A. x ph )  -> 
( -.  A. x ph  ->  -.  x  =  y ) )
32al2imi 1548 . . 3  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ( A. x  -.  A. x ph  ->  A. x  -.  x  =  y ) )
41, 3mtoi 169 . 2  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  -.  A. x  -.  A. x ph )
5 ax9lem4.a . . . 4  |-  -.  A. w  -.  w  =  x
6 ax9lem4.c . . . 4  |-  -.  A. x  -.  x  =  w
75, 6ax9lem3 29142 . . 3  |-  ( A. x ph  ->  ph )
8 hbn1 1704 . . 3  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
97, 8nsyl4 134 . 2  |-  ( -. 
A. x  -.  A. x ph  ->  ph )
104, 9syl 15 1  |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527
This theorem is referenced by:  ax9lem5  29144
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-6 1703  ax-11 1715
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