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Theorem lhpexle1lem 30818
Description: Lemma for lhpexle1 30819 and others that eliminates restrictions on  X. (Contributed by NM, 24-Jul-2013.)
Hypotheses
Ref Expression
lhpexle1lem.1  |-  ( ph  ->  E. p  e.  A  ( p  .<_  W  /\  ps ) )
lhpexle1lem.2  |-  ( (
ph  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X ) )
Assertion
Ref Expression
lhpexle1lem  |-  ( ph  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X
) )
Distinct variable groups:    .<_ , p    A, p    W, p    X, p    ph, p
Allowed substitution hint:    ps( p)

Proof of Theorem lhpexle1lem
StepHypRef Expression
1 lhpexle1lem.1 . . . 4  |-  ( ph  ->  E. p  e.  A  ( p  .<_  W  /\  ps ) )
21adantr 451 . . 3  |-  ( (
ph  /\  -.  X  e.  A )  ->  E. p  e.  A  ( p  .<_  W  /\  ps )
)
3 simprl 732 . . . . . 6  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  p  .<_  W )
4 simprr 733 . . . . . 6  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  ps )
5 simplr 731 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  p  e.  A )
6 simpllr 735 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  -.  X  e.  A )
7 nelne2 2549 . . . . . . 7  |-  ( ( p  e.  A  /\  -.  X  e.  A
)  ->  p  =/=  X )
85, 6, 7syl2anc 642 . . . . . 6  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  p  =/=  X )
93, 4, 83jca 1132 . . . . 5  |-  ( ( ( ( ph  /\  -.  X  e.  A
)  /\  p  e.  A )  /\  (
p  .<_  W  /\  ps ) )  ->  (
p  .<_  W  /\  ps  /\  p  =/=  X ) )
109ex 423 . . . 4  |-  ( ( ( ph  /\  -.  X  e.  A )  /\  p  e.  A
)  ->  ( (
p  .<_  W  /\  ps )  ->  ( p  .<_  W  /\  ps  /\  p  =/=  X ) ) )
1110reximdva 2668 . . 3  |-  ( (
ph  /\  -.  X  e.  A )  ->  ( E. p  e.  A  ( p  .<_  W  /\  ps )  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X ) ) )
122, 11mpd 14 . 2  |-  ( (
ph  /\  -.  X  e.  A )  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X ) )
131adantr 451 . . 3  |-  ( (
ph  /\  -.  X  .<_  W )  ->  E. p  e.  A  ( p  .<_  W  /\  ps )
)
14 simprl 732 . . . . . 6  |-  ( ( ( ph  /\  -.  X  .<_  W )  /\  ( p  .<_  W  /\  ps ) )  ->  p  .<_  W )
15 simprr 733 . . . . . 6  |-  ( ( ( ph  /\  -.  X  .<_  W )  /\  ( p  .<_  W  /\  ps ) )  ->  ps )
16 simplr 731 . . . . . . 7  |-  ( ( ( ph  /\  -.  X  .<_  W )  /\  ( p  .<_  W  /\  ps ) )  ->  -.  X  .<_  W )
17 nbrne2 4057 . . . . . . 7  |-  ( ( p  .<_  W  /\  -.  X  .<_  W )  ->  p  =/=  X
)
1814, 16, 17syl2anc 642 . . . . . 6  |-  ( ( ( ph  /\  -.  X  .<_  W )  /\  ( p  .<_  W  /\  ps ) )  ->  p  =/=  X )
1914, 15, 183jca 1132 . . . . 5  |-  ( ( ( ph  /\  -.  X  .<_  W )  /\  ( p  .<_  W  /\  ps ) )  ->  (
p  .<_  W  /\  ps  /\  p  =/=  X ) )
2019ex 423 . . . 4  |-  ( (
ph  /\  -.  X  .<_  W )  ->  (
( p  .<_  W  /\  ps )  ->  ( p 
.<_  W  /\  ps  /\  p  =/=  X ) ) )
2120reximdv 2667 . . 3  |-  ( (
ph  /\  -.  X  .<_  W )  ->  ( E. p  e.  A  ( p  .<_  W  /\  ps )  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X ) ) )
2213, 21mpd 14 . 2  |-  ( (
ph  /\  -.  X  .<_  W )  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X ) )
23 lhpexle1lem.2 . 2  |-  ( (
ph  /\  ( X  e.  A  /\  X  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X ) )
2412, 22, 23pm2.61dda 768 1  |-  ( ph  ->  E. p  e.  A  ( p  .<_  W  /\  ps  /\  p  =/=  X
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039
This theorem is referenced by:  lhpexle1  30819  lhpexle2  30821  lhpexle3  30823
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-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040
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