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Theorem reximddv2 2470
Description: Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Hypotheses
Ref Expression
reximddv2.1  |-  ( ( ( ( ph  /\  x  e.  A )  /\  y  e.  B
)  /\  ps )  ->  ch )
reximddv2.2  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )
Assertion
Ref Expression
reximddv2  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ch )
Distinct variable groups:    y, A    ph, x, y
Allowed substitution hints:    ps( x, y)    ch( x, y)    A( x)    B( x, y)

Proof of Theorem reximddv2
StepHypRef Expression
1 reximddv2.1 . . . . 5  |-  ( ( ( ( ph  /\  x  e.  A )  /\  y  e.  B
)  /\  ps )  ->  ch )
21ex 113 . . . 4  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  B )  ->  ( ps  ->  ch ) )
32reximdva 2468 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( E. y  e.  B  ps  ->  E. y  e.  B  ch ) )
43impr 371 . 2  |-  ( (
ph  /\  ( x  e.  A  /\  E. y  e.  B  ps )
)  ->  E. y  e.  B  ch )
5 reximddv2.2 . 2  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ps )
64, 5reximddv 2469 1  |-  ( ph  ->  E. x  e.  A  E. y  e.  B  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   E.wrex 2354
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-ral 2358  df-rex 2359
This theorem is referenced by: (None)
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