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Theorem ax9lem9 29148
Description: Lemma for ax9 1889. Similar to hbimd 1721, 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
ax9lem9.a  |-  -.  A. w  -.  w  =  x
ax9lem9.c  |-  -.  A. x  -.  x  =  w
ax9lem9.1  |-  ( ph  ->  A. x ph )
ax9lem9.2  |-  ( ph  ->  ( ps  ->  A. x ps ) )
ax9lem9.3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
Assertion
Ref Expression
ax9lem9  |-  ( ph  ->  ( ( ps  ->  ch )  ->  A. x
( ps  ->  ch ) ) )
Distinct variable groups:    x, w    ps, w
Allowed substitution hints:    ph( x, w)    ps( x)    ch( x, w)

Proof of Theorem ax9lem9
StepHypRef Expression
1 ax9lem9.1 . . . . 5  |-  ( ph  ->  A. x ph )
2 ax9lem9.2 . . . . 5  |-  ( ph  ->  ( ps  ->  A. x ps ) )
31, 2alrimih 1552 . . . 4  |-  ( ph  ->  A. x ( ps 
->  A. x ps )
)
4 ax9lem9.a . . . . . . 7  |-  -.  A. w  -.  w  =  x
5 ax9lem9.c . . . . . . 7  |-  -.  A. x  -.  x  =  w
64, 5ax9lem3 29142 . . . . . 6  |-  ( A. x ps  ->  ps )
7 hbn1 1704 . . . . . 6  |-  ( -. 
A. x ps  ->  A. x  -.  A. x ps )
86, 7nsyl4 134 . . . . 5  |-  ( -. 
A. x  -.  A. x ps  ->  ps )
98con1i 121 . . . 4  |-  ( -. 
ps  ->  A. x  -.  A. x ps )
10 con3 126 . . . . 5  |-  ( ( ps  ->  A. x ps )  ->  ( -. 
A. x ps  ->  -. 
ps ) )
1110al2imi 1548 . . . 4  |-  ( A. x ( ps  ->  A. x ps )  -> 
( A. x  -.  A. x ps  ->  A. x  -.  ps ) )
123, 9, 11syl2im 34 . . 3  |-  ( ph  ->  ( -.  ps  ->  A. x  -.  ps )
)
13 pm2.21 100 . . . 4  |-  ( -. 
ps  ->  ( ps  ->  ch ) )
1413alimi 1546 . . 3  |-  ( A. x  -.  ps  ->  A. x
( ps  ->  ch ) )
1512, 14syl6 29 . 2  |-  ( ph  ->  ( -.  ps  ->  A. x ( ps  ->  ch ) ) )
16 ax9lem9.3 . . 3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
17 ax-1 5 . . . 4  |-  ( ch 
->  ( ps  ->  ch ) )
1817alimi 1546 . . 3  |-  ( A. x ch  ->  A. x
( ps  ->  ch ) )
1916, 18syl6 29 . 2  |-  ( ph  ->  ( ch  ->  A. x
( ps  ->  ch ) ) )
2015, 19jad 154 1  |-  ( ph  ->  ( ( ps  ->  ch )  ->  A. x
( ps  ->  ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527
This theorem is referenced by:  ax9lem12  29151  ax9lem13  29152  ax9lem17  29156
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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