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Theorem bnj1345 34303
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1345.1  |-  ( ph  ->  E. x ( ps 
/\  ch ) )
bnj1345.2  |-  ( th  <->  (
ph  /\  ps  /\  ch ) )
bnj1345.3  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
bnj1345  |-  ( ph  ->  E. x th )

Proof of Theorem bnj1345
StepHypRef Expression
1 bnj1345.1 . . 3  |-  ( ph  ->  E. x ( ps 
/\  ch ) )
2 bnj1345.3 . . 3  |-  ( ph  ->  A. x ph )
31, 2bnj1275 34292 . 2  |-  ( ph  ->  E. x ( ph  /\ 
ps  /\  ch )
)
4 bnj1345.2 . 2  |-  ( th  <->  (
ph  /\  ps  /\  ch ) )
53, 4bnj1198 34274 1  |-  ( ph  ->  E. x th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971   A.wal 1396   E.wex 1617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-ex 1618  df-nf 1622
This theorem is referenced by:  bnj1379  34309  bnj1521  34329
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