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Theorem ax9lem11 29150
Description: Lemma for ax9 1889. Similar to exlimih 1729, 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
ax9lem11.a  |-  -.  A. w  -.  w  =  x
ax9lem11.c  |-  -.  A. x  -.  x  =  w
ax9lem11.1  |-  ( ps 
->  A. x ps )
ax9lem11.2  |-  ( ph  ->  ps )
Assertion
Ref Expression
ax9lem11  |-  ( E. x ph  ->  ps )
Distinct variable groups:    x, w    ps, w
Allowed substitution hints:    ph( x, w)    ps( x)

Proof of Theorem ax9lem11
StepHypRef Expression
1 ax9lem11.2 . . 3  |-  ( ph  ->  ps )
21eximi 1563 . 2  |-  ( E. x ph  ->  E. x ps )
3 df-ex 1529 . . 3  |-  ( E. x ps  <->  -.  A. x  -.  ps )
4 ax9lem11.a . . . . 5  |-  -.  A. w  -.  w  =  x
5 ax9lem11.c . . . . 5  |-  -.  A. x  -.  x  =  w
6 ax9lem11.1 . . . . 5  |-  ( ps 
->  A. x ps )
74, 5, 6ax9lem8 29147 . . . 4  |-  ( -. 
ps  ->  A. x  -.  ps )
87con1i 121 . . 3  |-  ( -. 
A. x  -.  ps  ->  ps )
93, 8sylbi 187 . 2  |-  ( E. x ps  ->  ps )
102, 9syl 15 1  |-  ( E. x ph  ->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527   E.wex 1528
This theorem is referenced by:  ax9lem15  29154
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
This theorem depends on definitions:  df-bi 177  df-ex 1529
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