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Theorem bnj596 33510
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj596.1  |-  ( ph  ->  A. x ph )
bnj596.2  |-  ( ph  ->  E. x ps )
Assertion
Ref Expression
bnj596  |-  ( ph  ->  E. x ( ph  /\ 
ps ) )

Proof of Theorem bnj596
StepHypRef Expression
1 bnj596.2 . . 3  |-  ( ph  ->  E. x ps )
21ancli 551 . 2  |-  ( ph  ->  ( ph  /\  E. x ps ) )
3 bnj596.1 . . . 4  |-  ( ph  ->  A. x ph )
43nfi 1608 . . 3  |-  F/ x ph
5419.42 1956 . 2  |-  ( E. x ( ph  /\  ps )  <->  ( ph  /\  E. x ps ) )
62, 5sylibr 212 1  |-  ( ph  ->  E. x ( ph  /\ 
ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1379   E.wex 1597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-12 1838
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1598  df-nf 1602
This theorem is referenced by:  bnj1275  33579  bnj1340  33589  bnj594  33677  bnj1398  33797
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