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Theorem ax9lem8 29769
Description: Lemma for ax9 1902. Similar to hbn 1732, without using sp 1728, ax9 1902, or ax10 1897. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ax9lem8.a  |-  -.  A. w  -.  w  =  x
ax9lem8.c  |-  -.  A. x  -.  x  =  w
ax9lem8.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
ax9lem8  |-  ( -. 
ph  ->  A. x  -.  ph )
Distinct variable groups:    x, w    ph, w
Allowed substitution hint:    ph( x)

Proof of Theorem ax9lem8
StepHypRef Expression
1 ax9lem8.a . . . 4  |-  -.  A. w  -.  w  =  x
2 ax9lem8.c . . . 4  |-  -.  A. x  -.  x  =  w
31, 2ax9lem3 29764 . . 3  |-  ( A. x ph  ->  ph )
43con3i 127 . 2  |-  ( -. 
ph  ->  -.  A. x ph )
5 hbn1 1716 . . 3  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
6 ax9lem8.1 . . . 4  |-  ( ph  ->  A. x ph )
76con3i 127 . . 3  |-  ( -. 
A. x ph  ->  -. 
ph )
85, 7alrimih 1555 . 2  |-  ( -. 
A. x ph  ->  A. x  -.  ph )
94, 8syl 15 1  |-  ( -. 
ph  ->  A. x  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530
This theorem is referenced by:  ax9lem10  29771  ax9lem11  29772  ax9lem12  29773  ax9lem13  29774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-8 1661  ax-6 1715  ax-11 1727
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