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Theorem 19.9d2r 24005
Description: A deduction version of one direction of 19.9 1800 with two variables (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypotheses
Ref Expression
19.9d2r.1  |-  ( ph  ->  F/ x ps )
19.9d2r.2  |-  ( ph  ->  F/ y ps )
19.9d2r.3  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )
Assertion
Ref Expression
19.9d2r  |-  ( ph  ->  ps )
Distinct variable group:    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)    A( x, y)    B( x, y)

Proof of Theorem 19.9d2r
StepHypRef Expression
1 nfv 1631 . 2  |-  F/ y
ph
2 19.9d2r.1 . 2  |-  ( ph  ->  F/ x ps )
3 19.9d2r.2 . 2  |-  ( ph  ->  F/ y ps )
4 19.9d2r.3 . 2  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )
51, 2, 3, 419.9d2rf 24004 1  |-  ( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1554   E.wrex 2713
This theorem is referenced by:  xrofsup  24161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555  df-rex 2718
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