MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.29af Structured version   Unicode version

Theorem r19.29af 2866
Description: A commonly used pattern based on r19.29 2862 (Contributed by Thierry Arnoux, 29-Nov-2017.)
Hypotheses
Ref Expression
r19.29af.0  |-  F/ x ph
r19.29af.1  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )
r19.29af.2  |-  ( ph  ->  E. x  e.  A  ps )
Assertion
Ref Expression
r19.29af  |-  ( ph  ->  ch )
Distinct variable group:    ch, x
Allowed substitution hints:    ph( x)    ps( x)    A( x)

Proof of Theorem r19.29af
StepHypRef Expression
1 r19.29af.0 . 2  |-  F/ x ph
2 nfv 1673 . 2  |-  F/ x ch
3 r19.29af.1 . 2  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )
4 r19.29af.2 . 2  |-  ( ph  ->  E. x  e.  A  ps )
51, 2, 3, 4r19.29af2 2865 1  |-  ( ph  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   F/wnf 1589    e. wcel 1756   E.wrex 2721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-12 1792
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-ral 2725  df-rex 2726
This theorem is referenced by:  r19.29a  2867  neiptopnei  18741  neitr  18789  utopsnneiplem  19827  isucn2  19859  restmetu  20167  colline  23057  f1otrg  23122  isarchi3  26209  ordtconlem1  26359  oms0  26715  eulerpartlemgvv  26764  stoweidlem27  29827  stoweidlem35  29835
  Copyright terms: Public domain W3C validator