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Theorem r19.29af2 2497
Description: A commonly used pattern based on r19.29 2495 (Contributed by Thierry Arnoux, 17-Dec-2017.)
Hypotheses
Ref Expression
r19.29af2.p  |-  F/ x ph
r19.29af2.c  |-  F/ x ch
r19.29af2.1  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )
r19.29af2.2  |-  ( ph  ->  E. x  e.  A  ps )
Assertion
Ref Expression
r19.29af2  |-  ( ph  ->  ch )

Proof of Theorem r19.29af2
StepHypRef Expression
1 r19.29af2.2 . . 3  |-  ( ph  ->  E. x  e.  A  ps )
2 r19.29af2.p . . . 4  |-  F/ x ph
3 r19.29af2.1 . . . . 5  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )
43exp31 356 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )
52, 4ralrimi 2433 . . 3  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
61, 5jca 300 . 2  |-  ( ph  ->  ( E. x  e.  A  ps  /\  A. x  e.  A  ( ps  ->  ch ) ) )
7 r19.29r 2496 . 2  |-  ( ( E. x  e.  A  ps  /\  A. x  e.  A  ( ps  ->  ch ) )  ->  E. x  e.  A  ( ps  /\  ( ps  ->  ch ) ) )
8 r19.29af2.c . . 3  |-  F/ x ch
9 pm3.35 339 . . . 4  |-  ( ( ps  /\  ( ps 
->  ch ) )  ->  ch )
109a1i 9 . . 3  |-  ( x  e.  A  ->  (
( ps  /\  ( ps  ->  ch ) )  ->  ch ) )
118, 10rexlimi 2471 . 2  |-  ( E. x  e.  A  ( ps  /\  ( ps 
->  ch ) )  ->  ch )
126, 7, 113syl 17 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   F/wnf 1390    e. wcel 1434   A.wral 2349   E.wrex 2350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-ral 2354  df-rex 2355
This theorem is referenced by:  r19.29af  2498
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