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Theorem r19.29a 2607
Description: A commonly used pattern based on r19.29 2601. (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 1515 . 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 2605 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 2135   E.wrex 2443
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 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-17 1513  ax-ial 1521  ax-i5r 1522
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-ral 2447  df-rex 2448
This theorem is referenced by:  cnegexlem3  8069  cnegex  8070  modqmuladdnn0  10297  uzwodc  11964  1arith  12291  neitx  12866
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