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Theorem r19.29d2r 2621
Description: Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypotheses
Ref Expression
r19.29d2r.1  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ps )
r19.29d2r.2  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ch )
Assertion
Ref Expression
r19.29d2r  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ( ps  /\  ch )
)

Proof of Theorem r19.29d2r
StepHypRef Expression
1 r19.29d2r.1 . . 3  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ps )
2 r19.29d2r.2 . . 3  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ch )
3 r19.29 2614 . . 3  |-  ( ( A. x  e.  A  A. y  e.  B  ps  /\  E. x  e.  A  E. y  e.  B  ch )  ->  E. x  e.  A  ( A. y  e.  B  ps  /\  E. y  e.  B  ch ) )
41, 2, 3syl2anc 411 . 2  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  ps  /\  E. y  e.  B  ch ) )
5 r19.29 2614 . . 3  |-  ( ( A. y  e.  B  ps  /\  E. y  e.  B  ch )  ->  E. y  e.  B  ( ps  /\  ch )
)
65reximi 2574 . 2  |-  ( E. x  e.  A  ( A. y  e.  B  ps  /\  E. y  e.  B  ch )  ->  E. x  e.  A  E. y  e.  B  ( ps  /\  ch )
)
74, 6syl 14 1  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ( ps  /\  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wral 2455   E.wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-ral 2460  df-rex 2461
This theorem is referenced by:  r19.29vva  2622  cauappcvgprlemdisj  7649
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