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Theorem bnj1459 28875
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1459.1  |-  ( ps  <->  (
ph  /\  x  e.  A ) )
bnj1459.2  |-  ( ps 
->  ch )
Assertion
Ref Expression
bnj1459  |-  ( ph  ->  A. x  e.  A  ch )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)

Proof of Theorem bnj1459
StepHypRef Expression
1 bnj1459.1 . . 3  |-  ( ps  <->  (
ph  /\  x  e.  A ) )
2 bnj1459.2 . . 3  |-  ( ps 
->  ch )
31, 2sylbir 204 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ch )
43ralrimiva 2626 1  |-  ( ph  ->  A. x  e.  A  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   A.wral 2543
This theorem is referenced by:  bnj1501  29097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-nf 1532  df-ral 2548
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