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Theorem r19.29af 2576
Description: A commonly used pattern based on r19.29 2572 (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 1509 . 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 2575 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   F/wnf 1437    e. wcel 1481   E.wrex 2418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-i5r 1516
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-ral 2422  df-rex 2423
This theorem is referenced by:  r19.29an  2577  r19.29a  2578  suplocsrlem  7640  supinfneg  9417  infsupneg  9418  pw1nct  13371
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