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Theorem ralrimdvva 2447
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008.)
Hypothesis
Ref Expression
ralrimdvva.1  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( ps  ->  ch ) )
Assertion
Ref Expression
ralrimdvva  |-  ( 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 ralrimdvva
StepHypRef Expression
1 ralrimdvva.1 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( ps  ->  ch ) )
21ex 113 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ( ps  ->  ch ) ) )
32com23 77 . 2  |-  ( ph  ->  ( ps  ->  (
( x  e.  A  /\  y  e.  B
)  ->  ch )
) )
43ralrimdvv 2446 1  |-  ( ph  ->  ( ps  ->  A. x  e.  A  A. y  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   A.wral 2349
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-4 1441  ax-17 1460
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-ral 2354
This theorem is referenced by:  isosolem  5494  isotilem  6478
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