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Theorem reximdv2 2506
Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.)
Hypothesis
Ref Expression
reximdv2.1  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  ->  ( x  e.  B  /\  ch ) ) )
Assertion
Ref Expression
reximdv2  |-  ( ph  ->  ( E. x  e.  A  ps  ->  E. x  e.  B  ch )
)
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)    B( x)

Proof of Theorem reximdv2
StepHypRef Expression
1 reximdv2.1 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  ->  ( x  e.  B  /\  ch ) ) )
21eximdv 1834 . 2  |-  ( ph  ->  ( E. x ( x  e.  A  /\  ps )  ->  E. x
( x  e.  B  /\  ch ) ) )
3 df-rex 2397 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
4 df-rex 2397 . 2  |-  ( E. x  e.  B  ch  <->  E. x ( x  e.  B  /\  ch )
)
52, 3, 43imtr4g 204 1  |-  ( ph  ->  ( E. x  e.  A  ps  ->  E. x  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1451    e. wcel 1463   E.wrex 2392
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  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-rex 2397
This theorem is referenced by:  reximssdv  2511  ssrexv  3130  ssimaex  5448  ico0  9990  ioc0  9991  r19.2uz  10716  trilpolemlt1  13068
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