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Theorem bnj1521 28394
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1521.1  |-  ( ch 
->  E. x  e.  B  ph )
bnj1521.2  |-  ( th  <->  ( ch  /\  x  e.  B  /\  ph )
)
bnj1521.3  |-  ( ch 
->  A. x ch )
Assertion
Ref Expression
bnj1521  |-  ( ch 
->  E. x th )

Proof of Theorem bnj1521
StepHypRef Expression
1 bnj1521.1 . . 3  |-  ( ch 
->  E. x  e.  B  ph )
21bnj1196 28338 . 2  |-  ( ch 
->  E. x ( x  e.  B  /\  ph ) )
3 bnj1521.2 . 2  |-  ( th  <->  ( ch  /\  x  e.  B  /\  ph )
)
4 bnj1521.3 . 2  |-  ( ch 
->  A. x ch )
52, 3, 4bnj1345 28368 1  |-  ( ch 
->  E. x th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934   A.wal 1531   E.wex 1532    e. wcel 1701   E.wrex 2578
This theorem is referenced by:  bnj1204  28553  bnj1311  28565  bnj1398  28575  bnj1408  28577  bnj1450  28591  bnj1312  28599  bnj1501  28608  bnj1523  28612
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-11 1732
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1533  df-nf 1536  df-rex 2583
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