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Theorem r19.29_2a 2862
Description: A commonly used pattern based on r19.29 2855, version with two restricted quantifiers (Contributed by Thierry Arnoux, 26-Nov-2017.)
Hypotheses
Ref Expression
r19.29_2a.1  |-  ( ( ( ( ph  /\  x  e.  A )  /\  y  e.  B
)  /\  ps )  ->  ch )
r19.29_2a.2  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )
Assertion
Ref Expression
r19.29_2a  |-  ( ph  ->  ch )
Distinct variable groups:    y, A    x, y, ch    ph, x, y
Allowed substitution hints:    ps( x, y)    A( x)    B( x, y)

Proof of Theorem r19.29_2a
StepHypRef Expression
1 r19.29_2a.1 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  A )  /\  y  e.  B
)  /\  ps )  ->  ch )
21ex 434 . . . . 5  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  B )  ->  ( ps  ->  ch ) )
32ralrimiva 2797 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  A. y  e.  B  ( ps  ->  ch ) )
43ralrimiva 2797 . . 3  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ( ps  ->  ch )
)
5 r19.29_2a.2 . . 3  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )
64, 5r19.29d2r 2861 . 2  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ( ( ps  ->  ch )  /\  ps )
)
7 pm3.35 584 . . . . 5  |-  ( ( ps  /\  ( ps 
->  ch ) )  ->  ch )
87ancoms 450 . . . 4  |-  ( ( ( ps  ->  ch )  /\  ps )  ->  ch )
98rexlimivw 2835 . . 3  |-  ( E. y  e.  B  ( ( ps  ->  ch )  /\  ps )  ->  ch )
109rexlimivw 2835 . 2  |-  ( E. x  e.  A  E. y  e.  B  (
( ps  ->  ch )  /\  ps )  ->  ch )
116, 10syl 16 1  |-  ( ph  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1761   A.wral 2713   E.wrex 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-12 1797
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-nf 1595  df-ral 2718  df-rex 2719
This theorem is referenced by:  trust  19763  utoptop  19768  metusttoOLD  20091  metustto  20092  restmetu  20121  tgbtwndiff  22918  legov  22971  tglinethru  22996  f1otrge  23053  archiabllem2c  26145  pstmfval  26259  eulerpartlemgvv  26689
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