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Theorem r19.29af2 2942
Description: A commonly used pattern based on r19.29 2939. (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 602 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )
62, 3, 5rexlimd 2885 . 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 367   F/wnf 1635    e. wcel 1840   E.wrex 2752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-12 1876
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1632  df-nf 1636  df-ral 2756  df-rex 2757
This theorem is referenced by:  r19.29af  2944  restmetu  21272  aciunf1lem  27827  locfinreflem  28177  esumrnmpt2  28396  esum2dlem  28420  esum2d  28421  esumiun  28422
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