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Theorem r19.29a 2454
Description: A commonly used pattern based on r19.29 2450 (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 1421 . 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 2453 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    e. wcel 1393   E.wrex 2307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-ral 2311  df-rex 2312
This theorem is referenced by:  cnegexlem3  7188  cnegex  7189
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