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Theorem ax9lem7 29768
Description: Lemma for ax9 1902. Similar to hba1 1731, 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
ax9lem7.a  |-  -.  A. w  -.  w  =  x
ax9lem7.c  |-  -.  A. x  -.  x  =  w
Assertion
Ref Expression
ax9lem7  |-  ( A. x ph  ->  A. x A. x ph )
Distinct variable groups:    x, w    ph, w
Allowed substitution hint:    ph( x)

Proof of Theorem ax9lem7
StepHypRef Expression
1 ax9lem7.a . . . 4  |-  -.  A. w  -.  w  =  x
2 ax9lem7.c . . . 4  |-  -.  A. x  -.  x  =  w
31, 2ax9lem3 29764 . . 3  |-  ( A. x  -.  A. x ph  ->  -.  A. x ph )
43con2i 112 . 2  |-  ( A. x ph  ->  -.  A. x  -.  A. x ph )
5 hbn1 1716 . 2  |-  ( -. 
A. x  -.  A. x ph  ->  A. x  -.  A. x  -.  A. x ph )
6 hbn1 1716 . . . 4  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
76con1i 121 . . 3  |-  ( -. 
A. x  -.  A. x ph  ->  A. x ph )
87alimi 1549 . 2  |-  ( A. x  -.  A. x  -.  A. x ph  ->  A. x A. x ph )
94, 5, 83syl 18 1  |-  ( A. x ph  ->  A. x A. x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530
This theorem is referenced by:  ax9lem12  29773  ax9lem13  29774  ax9vax9  29780
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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