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Theorem 19.9d2rf 25867
Description: A deduction version of one direction of 19.9 1827 with two variables (Contributed by Thierry Arnoux, 20-Mar-2017.)
Hypotheses
Ref Expression
19.9d2rf.0  |-  F/ y
ph
19.9d2rf.1  |-  ( ph  ->  F/ x ps )
19.9d2rf.2  |-  ( ph  ->  F/ y ps )
19.9d2rf.3  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )
Assertion
Ref Expression
19.9d2rf  |-  ( ph  ->  ps )

Proof of Theorem 19.9d2rf
StepHypRef Expression
1 19.9d2rf.3 . . . 4  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )
2 rexex 2780 . . . 4  |-  ( E. x  e.  A  E. y  e.  B  ps  ->  E. x E. y  e.  B  ps )
3 rexex 2780 . . . . 5  |-  ( E. y  e.  B  ps  ->  E. y ps )
43eximi 1625 . . . 4  |-  ( E. x E. y  e.  B  ps  ->  E. x E. y ps )
51, 2, 43syl 20 . . 3  |-  ( ph  ->  E. x E. y ps )
6 19.9d2rf.0 . . . . 5  |-  F/ y
ph
7 19.9d2rf.1 . . . . 5  |-  ( ph  ->  F/ x ps )
86, 7nfexd 1878 . . . 4  |-  ( ph  ->  F/ x E. y ps )
9819.9d 1826 . . 3  |-  ( ph  ->  ( E. x E. y ps  ->  E. y ps ) )
105, 9mpd 15 . 2  |-  ( ph  ->  E. y ps )
11 19.9d2rf.2 . . 3  |-  ( ph  ->  F/ y ps )
121119.9d 1826 . 2  |-  ( ph  ->  ( E. y ps 
->  ps ) )
1310, 12mpd 15 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wex 1586   F/wnf 1589   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-11 1780  ax-12 1792
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590  df-rex 2726
This theorem is referenced by:  19.9d2r  25868  xrofsup  26060
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