MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralrimdvv Unicode version

Theorem ralrimdvv 2639
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 418 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ch ) )
32ralrimivv 2636 . 2  |-  ( (
ph  /\  ps )  ->  A. x  e.  A  A. y  e.  B  ch )
43ex 423 1  |-  ( ph  ->  ( ps  ->  A. x  e.  A  A. y  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1686   A.wral 2545
This theorem is referenced by:  ralrimdvva  2640  clatl  14222  lspsneu  15878  aalioulem4  19717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-11 1717
This theorem depends on definitions:  df-bi 177  df-an 360  df-nf 1534  df-ral 2550
  Copyright terms: Public domain W3C validator