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Theorem r19.29af 2498
Description: A commonly used pattern based on r19.29 2495 (Contributed by Thierry Arnoux, 29-Nov-2017.)
Hypotheses
Ref Expression
r19.29af.0  |-  F/ x ph
r19.29af.1  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )
r19.29af.2  |-  ( ph  ->  E. x  e.  A  ps )
Assertion
Ref Expression
r19.29af  |-  ( ph  ->  ch )
Distinct variable group:    ch, x
Allowed substitution hints:    ph( x)    ps( x)    A( x)

Proof of Theorem r19.29af
StepHypRef Expression
1 r19.29af.0 . 2  |-  F/ x ph
2 nfv 1462 . 2  |-  F/ x ch
3 r19.29af.1 . 2  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )
4 r19.29af.2 . 2  |-  ( ph  ->  E. x  e.  A  ps )
51, 2, 3, 4r19.29af2 2497 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   F/wnf 1390    e. wcel 1434   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:  r19.29a  2499  supinfneg  8764  infsupneg  8765
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