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Theorem reximssdv 2539
Description: Derivation of a restricted existential quantification over a subset (the second hypothesis implies  A  C_  B), deduction form. (Contributed by AV, 21-Aug-2022.)
Hypotheses
Ref Expression
reximssdv.1  |-  ( ph  ->  E. x  e.  B  ps )
reximssdv.2  |-  ( (
ph  /\  ( x  e.  B  /\  ps )
)  ->  x  e.  A )
reximssdv.3  |-  ( (
ph  /\  ( x  e.  B  /\  ps )
)  ->  ch )
Assertion
Ref Expression
reximssdv  |-  ( ph  ->  E. x  e.  A  ch )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)    B( x)

Proof of Theorem reximssdv
StepHypRef Expression
1 reximssdv.1 . 2  |-  ( ph  ->  E. x  e.  B  ps )
2 reximssdv.2 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  ps )
)  ->  x  e.  A )
3 reximssdv.3 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  ps )
)  ->  ch )
42, 3jca 304 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  ps )
)  ->  ( x  e.  A  /\  ch )
)
54ex 114 . . 3  |-  ( ph  ->  ( ( x  e.  B  /\  ps )  ->  ( x  e.  A  /\  ch ) ) )
65reximdv2 2534 . 2  |-  ( ph  ->  ( E. x  e.  B  ps  ->  E. x  e.  A  ch )
)
71, 6mpd 13 1  |-  ( ph  ->  E. x  e.  A  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1481   E.wrex 2418
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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-rex 2423
This theorem is referenced by:  suplocexprlemrl  7548  neissex  12371  iscnp4  12424  suplociccex  12809
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