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Theorem bnj1345 32120
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 32109 . 2  |-  ( ph  ->  E. x ( ph  /\ 
ps  /\  ch )
)
4 bnj1345.2 . 2  |-  ( th  <->  (
ph  /\  ps  /\  ch ) )
53, 4bnj1198 32091 1  |-  ( ph  ->  E. x th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1368   E.wex 1587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-ex 1588  df-nf 1591
This theorem is referenced by:  bnj1379  32126  bnj1521  32146
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