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Theorem ralrimdvv 2866
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)
Hypothesis
Ref Expression
ralrimdvv.1  |-  ( ph  ->  ( ps  ->  (
( x  e.  A  /\  y  e.  B
)  ->  ch )
) )
Assertion
Ref Expression
ralrimdvv  |-  ( ph  ->  ( ps  ->  A. x  e.  A  A. y  e.  B  ch )
)
Distinct variable groups:    x, y, ph    ps, x, y    y, A
Allowed substitution hints:    ch( x, y)    A( x)    B( x, y)

Proof of Theorem ralrimdvv
StepHypRef Expression
1 ralrimdvv.1 . . . 4  |-  ( ph  ->  ( ps  ->  (
( x  e.  A  /\  y  e.  B
)  ->  ch )
) )
21imp 429 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ch ) )
32ralrimivv 2863 . 2  |-  ( (
ph  /\  ps )  ->  A. x  e.  A  A. y  e.  B  ch )
43ex 434 1  |-  ( ph  ->  ( ps  ->  A. x  e.  A  A. y  e.  B  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1804   A.wral 2793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691
This theorem depends on definitions:  df-bi 185  df-an 371  df-ral 2798
This theorem is referenced by:  ralrimdvva  2867  lspsneu  17748  pmatcoe1fsupp  19180  aalioulem4  22709  fargshiftf1  24615
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