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Theorem ax9lem12 29151
Description: Lemma for ax9 1889. Similar to spime 1916 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
ax9lem12.a  |-  -.  A. w  -.  w  =  x
ax9lem12.c  |-  -.  A. x  -.  x  =  w
ax9lem12.d  |-  -.  A. x  -.  x  =  y
ax9lem12.1  |-  ( x  =  y  ->  ( ph  ->  ps ) )
ax9lem12.2  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
ax9lem12  |-  ( ph  ->  E. x ps )
Distinct variable groups:    x, y    x, w    ph, w    ps, w
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem ax9lem12
StepHypRef Expression
1 ax9lem12.d . . . 4  |-  -.  A. x  -.  x  =  y
2 ax9lem12.a . . . . . 6  |-  -.  A. w  -.  w  =  x
3 ax9lem12.c . . . . . 6  |-  -.  A. x  -.  x  =  w
4 id 19 . . . . . . 7  |-  ( ph  ->  ph )
54hbth 1539 . . . . . . . 8  |-  ( (
ph  ->  ph )  ->  A. x
( ph  ->  ph )
)
62, 3ax9lem7 29146 . . . . . . . . 9  |-  ( A. x  -.  ps  ->  A. x A. x  -.  ps )
76a1i 10 . . . . . . . 8  |-  ( (
ph  ->  ph )  ->  ( A. x  -.  ps  ->  A. x A. x  -.  ps ) )
8 ax9lem12.2 . . . . . . . . . 10  |-  ( ph  ->  A. x ph )
92, 3, 8ax9lem8 29147 . . . . . . . . 9  |-  ( -. 
ph  ->  A. x  -.  ph )
109a1i 10 . . . . . . . 8  |-  ( (
ph  ->  ph )  ->  ( -.  ph  ->  A. x  -.  ph ) )
112, 3, 5, 7, 10ax9lem9 29148 . . . . . . 7  |-  ( (
ph  ->  ph )  ->  (
( A. x  -.  ps  ->  -.  ph )  ->  A. x ( A. x  -.  ps  ->  -.  ph )
) )
124, 11ax-mp 8 . . . . . 6  |-  ( ( A. x  -.  ps  ->  -.  ph )  ->  A. x ( A. x  -.  ps  ->  -.  ph )
)
132, 3, 12ax9lem8 29147 . . . . 5  |-  ( -.  ( A. x  -.  ps  ->  -.  ph )  ->  A. x  -.  ( A. x  -.  ps  ->  -. 
ph ) )
14 ax9lem12.1 . . . . . . 7  |-  ( x  =  y  ->  ( ph  ->  ps ) )
152, 3ax9lem3 29142 . . . . . . 7  |-  ( A. x  -.  ps  ->  -.  ps )
1614, 15nsyli 133 . . . . . 6  |-  ( x  =  y  ->  ( A. x  -.  ps  ->  -. 
ph ) )
1716con3i 127 . . . . 5  |-  ( -.  ( A. x  -.  ps  ->  -.  ph )  ->  -.  x  =  y
)
1813, 17alrimih 1552 . . . 4  |-  ( -.  ( A. x  -.  ps  ->  -.  ph )  ->  A. x  -.  x  =  y )
191, 18mt3 171 . . 3  |-  ( A. x  -.  ps  ->  -.  ph )
2019con2i 112 . 2  |-  ( ph  ->  -.  A. x  -.  ps )
21 df-ex 1529 . 2  |-  ( E. x ps  <->  -.  A. x  -.  ps )
2220, 21sylibr 203 1  |-  ( ph  ->  E. x 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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