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Theorem ralimdv2 2406
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)
Hypothesis
Ref Expression
ralimdv2.1  |-  ( ph  ->  ( ( x  e.  A  ->  ps )  ->  ( x  e.  B  ->  ch ) ) )
Assertion
Ref Expression
ralimdv2  |-  ( ph  ->  ( A. x  e.  A  ps  ->  A. x  e.  B  ch )
)
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)    B( x)

Proof of Theorem ralimdv2
StepHypRef Expression
1 ralimdv2.1 . . 3  |-  ( ph  ->  ( ( x  e.  A  ->  ps )  ->  ( x  e.  B  ->  ch ) ) )
21alimdv 1775 . 2  |-  ( ph  ->  ( A. x ( x  e.  A  ->  ps )  ->  A. x
( x  e.  B  ->  ch ) ) )
3 df-ral 2328 . 2  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
4 df-ral 2328 . 2  |-  ( A. x  e.  B  ch  <->  A. x ( x  e.  B  ->  ch )
)
52, 3, 43imtr4g 198 1  |-  ( ph  ->  ( A. x  e.  A  ps  ->  A. x  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257    e. wcel 1409   A.wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-17 1435
This theorem depends on definitions:  df-bi 114  df-ral 2328
This theorem is referenced by:  ssralv  3032  r19.29uz  9819
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