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Theorem ax9lem10 29149
Description: Lemma for ax9 1889. Similar to hban 1736, 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
ax9lem10.a  |-  -.  A. w  -.  w  =  x
ax9lem10.c  |-  -.  A. x  -.  x  =  w
ax9lem10.1  |-  ( ph  ->  A. x ph )
ax9lem10.2  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
ax9lem10  |-  ( (
ph  /\  ps )  ->  A. x ( ph  /\ 
ps ) )
Distinct variable groups:    x, w    ph, w    ps, w
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem ax9lem10
StepHypRef Expression
1 df-an 360 . 2  |-  ( (
ph  /\  ps )  <->  -.  ( ph  ->  -.  ps ) )
2 ax9lem10.a . . 3  |-  -.  A. w  -.  w  =  x
3 ax9lem10.c . . 3  |-  -.  A. x  -.  x  =  w
4 ax9lem10.1 . . . . . 6  |-  ( ph  ->  A. x ph )
52, 3, 4ax9lem8 29147 . . . . 5  |-  ( -. 
ph  ->  A. x  -.  ph )
6 pm2.21 100 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  -. 
ps ) )
75, 6alrimih 1552 . . . 4  |-  ( -. 
ph  ->  A. x ( ph  ->  -.  ps ) )
8 ax9lem10.2 . . . . . 6  |-  ( ps 
->  A. x ps )
92, 3, 8ax9lem8 29147 . . . . 5  |-  ( -. 
ps  ->  A. x  -.  ps )
10 ax-1 5 . . . . 5  |-  ( -. 
ps  ->  ( ph  ->  -. 
ps ) )
119, 10alrimih 1552 . . . 4  |-  ( -. 
ps  ->  A. x ( ph  ->  -.  ps ) )
127, 11ja 153 . . 3  |-  ( (
ph  ->  -.  ps )  ->  A. x ( ph  ->  -.  ps ) )
132, 3, 12ax9lem8 29147 . 2  |-  ( -.  ( ph  ->  -.  ps )  ->  A. x  -.  ( ph  ->  -.  ps ) )
141, 13hbxfrbi 1555 1  |-  ( (
ph  /\  ps )  ->  A. x ( ph  /\ 
ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527
This theorem is referenced by:  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
This theorem depends on definitions:  df-bi 177  df-an 360
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