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Theorem r19.29a 2633
Description: A commonly used pattern based on r19.29 2627. (Contributed by Thierry Arnoux, 22-Nov-2017.)
Hypotheses
Ref Expression
r19.29a.1  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )
r19.29a.2  |-  ( ph  ->  E. x  e.  A  ps )
Assertion
Ref Expression
r19.29a  |-  ( ph  ->  ch )
Distinct variable groups:    ch, x    ph, x
Allowed substitution hints:    ps( x)    A( x)

Proof of Theorem r19.29a
StepHypRef Expression
1 nfv 1539 . 2  |-  F/ x ph
2 r19.29a.1 . 2  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )
3 r19.29a.2 . 2  |-  ( ph  ->  E. x  e.  A  ps )
41, 2, 3r19.29af 2631 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2160   E.wrex 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-ral 2473  df-rex 2474
This theorem is referenced by:  cnegexlem3  8163  cnegex  8164  modqmuladdnn0  10398  uzwodc  12069  1arith  12398  mhmid  13054  mhmmnd  13055  ghmgrp  13057  ghmcmn  13262  ringinvnz1ne0  13398  neitx  14220
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