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Theorem r19.29vva 2513
Description: A commonly used pattern based on r19.29 2506, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
Hypotheses
Ref Expression
r19.29vva.1  |-  ( ( ( ( ph  /\  x  e.  A )  /\  y  e.  B
)  /\  ps )  ->  ch )
r19.29vva.2  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )
Assertion
Ref Expression
r19.29vva  |-  ( 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 r19.29vva
StepHypRef Expression
1 r19.29vva.1 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  A )  /\  y  e.  B
)  /\  ps )  ->  ch )
21ex 113 . . . . 5  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  B )  ->  ( ps  ->  ch ) )
32ralrimiva 2446 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  A. y  e.  B  ( ps  ->  ch ) )
43ralrimiva 2446 . . 3  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ( ps  ->  ch )
)
5 r19.29vva.2 . . 3  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )
64, 5r19.29d2r 2512 . 2  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ( ( ps  ->  ch )  /\  ps )
)
7 pm3.35 339 . . . . 5  |-  ( ( ps  /\  ( ps 
->  ch ) )  ->  ch )
87ancoms 264 . . . 4  |-  ( ( ( ps  ->  ch )  /\  ps )  ->  ch )
98rexlimivw 2485 . . 3  |-  ( E. y  e.  B  ( ( ps  ->  ch )  /\  ps )  ->  ch )
109rexlimivw 2485 . 2  |-  ( E. x  e.  A  E. y  e.  B  (
( ps  ->  ch )  /\  ps )  ->  ch )
116, 10syl 14 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1438   A.wral 2359   E.wrex 2360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-ral 2364  df-rex 2365
This theorem is referenced by: (None)
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