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Theorem ralimi2 2467
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)
Hypothesis
Ref Expression
ralimi2.1  |-  ( ( x  e.  A  ->  ph )  ->  ( x  e.  B  ->  ps ) )
Assertion
Ref Expression
ralimi2  |-  ( A. x  e.  A  ph  ->  A. x  e.  B  ps )

Proof of Theorem ralimi2
StepHypRef Expression
1 ralimi2.1 . . 3  |-  ( ( x  e.  A  ->  ph )  ->  ( x  e.  B  ->  ps ) )
21alimi 1414 . 2  |-  ( A. x ( x  e.  A  ->  ph )  ->  A. x ( x  e.  B  ->  ps )
)
3 df-ral 2396 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
4 df-ral 2396 . 2  |-  ( A. x  e.  B  ps  <->  A. x ( x  e.  B  ->  ps )
)
52, 3, 43imtr4i 200 1  |-  ( A. x  e.  A  ph  ->  A. x  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1312    e. wcel 1463   A.wral 2391
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 1406  ax-gen 1408
This theorem depends on definitions:  df-bi 116  df-ral 2396
This theorem is referenced by:  ralimia  2468  ralcom3  2573  bj-nntrans  12960  bj-findis  12988
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