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Theorem r19.29d2r 2997
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 2989 . . 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 659 . 2  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  ps  /\  E. y  e.  B  ch ) )
5 r19.29 2989 . . 3  |-  ( ( A. y  e.  B  ps  /\  E. y  e.  B  ch )  ->  E. y  e.  B  ( ps  /\  ch )
)
65reximi 2922 . 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 16 1  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ( ps  /\  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wral 2804   E.wrex 2805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-ral 2809  df-rex 2810
This theorem is referenced by:  r19.29vva  2998  ucnima  20950  tgisline  24208  rnmpt2ss  27742  xrofsup  27816
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