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Theorem r19.29af2 2981
Description: A commonly used pattern based on r19.29 2978. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Proof shortened by OpenAI, 25-Mar-2020.)
Hypotheses
Ref Expression
r19.29af2.p  |-  F/ x ph
r19.29af2.c  |-  F/ x ch
r19.29af2.1  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )
r19.29af2.2  |-  ( ph  ->  E. x  e.  A  ps )
Assertion
Ref Expression
r19.29af2  |-  ( ph  ->  ch )

Proof of Theorem r19.29af2
StepHypRef Expression
1 r19.29af2.2 . 2  |-  ( ph  ->  E. x  e.  A  ps )
2 r19.29af2.p . . 3  |-  F/ x ph
3 r19.29af2.c . . 3  |-  F/ x ch
4 r19.29af2.1 . . . 4  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )
54exp31 604 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )
62, 3, 5rexlimd 2927 . 2  |-  ( ph  ->  ( E. x  e.  A  ps  ->  ch ) )
71, 6mpd 15 1  |-  ( ph  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   F/wnf 1603    e. wcel 1804   E.wrex 2794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-12 1840
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1600  df-nf 1604  df-ral 2798  df-rex 2799
This theorem is referenced by:  r19.29af  2983  restmetu  21067  locfinreflem  27820
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