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Theorem ax9lem5 29144
Description: Lemma for ax9 1889. Similar to spim 1915 with distinct variables, 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
ax9lem5.a  |-  -.  A. w  -.  w  =  x
ax9lem5.c  |-  -.  A. x  -.  x  =  w
ax9lem5.d  |-  -.  A. x  -.  x  =  y
ax9lem5.2  |-  ( ps 
->  A. x ps )
ax9lem5.3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
ax9lem5  |-  ( A. x ph  ->  ps )
Distinct variable groups:    x, w    ps, w    x, y
Allowed substitution hints:    ph( x, y, w)    ps( x, y)

Proof of Theorem ax9lem5
StepHypRef Expression
1 ax9lem5.3 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  ps ) )
21com12 27 . . . 4  |-  ( ph  ->  ( x  =  y  ->  ps ) )
3 ax9lem5.2 . . . 4  |-  ( ps 
->  A. x ps )
42, 3syl6 29 . . 3  |-  ( ph  ->  ( x  =  y  ->  A. x ps )
)
54alimi 1546 . 2  |-  ( A. x ph  ->  A. x
( x  =  y  ->  A. x ps )
)
6 ax9lem5.a . . 3  |-  -.  A. w  -.  w  =  x
7 ax9lem5.c . . 3  |-  -.  A. x  -.  x  =  w
8 ax9lem5.d . . 3  |-  -.  A. x  -.  x  =  y
96, 7, 8ax9lem4 29143 . 2  |-  ( A. x ( x  =  y  ->  A. x ps )  ->  ps )
105, 9syl 15 1  |-  ( A. x ph  ->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527
This theorem is referenced by:  ax9lem6  29145
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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