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Theorem ralimdvva 2501
Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1433). (Contributed by AV, 27-Nov-2019.)
Hypothesis
Ref Expression
ralimdvva.1  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( ps  ->  ch ) )
Assertion
Ref Expression
ralimdvva  |-  ( ph  ->  ( A. x  e.  A  A. y  e.  B  ps  ->  A. x  e.  A  A. y  e.  B  ch )
)
Distinct variable groups:    y, A    x, y, ph
Allowed substitution hints:    ps( x, y)    ch( x, y)    A( x)    B( x, y)

Proof of Theorem ralimdvva
StepHypRef Expression
1 ralimdvva.1 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( ps  ->  ch ) )
21anassrs 397 . . 3  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  B )  ->  ( ps  ->  ch ) )
32ralimdva 2499 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( A. y  e.  B  ps  ->  A. y  e.  B  ch ) )
43ralimdva 2499 1  |-  ( ph  ->  ( A. x  e.  A  A. y  e.  B  ps  ->  A. x  e.  A  A. y  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1480   A.wral 2416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2421
This theorem is referenced by:  addcncntoplem  12730
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