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Theorem bnj1209 29105
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1209.1  |-  ( ch 
->  E. x  e.  B  ph )
bnj1209.2  |-  ( th  <->  ( ch  /\  x  e.  B  /\  ph )
)
Assertion
Ref Expression
bnj1209  |-  ( ch 
->  E. x th )
Distinct variable group:    ch, x
Allowed substitution hints:    ph( x)    th( x)    B( x)

Proof of Theorem bnj1209
StepHypRef Expression
1 bnj1209.1 . . . . 5  |-  ( ch 
->  E. x  e.  B  ph )
21bnj1196 29103 . . . 4  |-  ( ch 
->  E. x ( x  e.  B  /\  ph ) )
32ancli 535 . . 3  |-  ( ch 
->  ( ch  /\  E. x ( x  e.  B  /\  ph )
) )
4 19.42v 1928 . . 3  |-  ( E. x ( ch  /\  ( x  e.  B  /\  ph ) )  <->  ( ch  /\ 
E. x ( x  e.  B  /\  ph ) ) )
53, 4sylibr 204 . 2  |-  ( ch 
->  E. x ( ch 
/\  ( x  e.  B  /\  ph )
) )
6 bnj1209.2 . . 3  |-  ( th  <->  ( ch  /\  x  e.  B  /\  ph )
)
7 3anass 940 . . 3  |-  ( ( ch  /\  x  e.  B  /\  ph )  <->  ( ch  /\  ( x  e.  B  /\  ph ) ) )
86, 7bitri 241 . 2  |-  ( th  <->  ( ch  /\  ( x  e.  B  /\  ph ) ) )
95, 8bnj1198 29104 1  |-  ( ch 
->  E. x th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    e. wcel 1725   E.wrex 2698
This theorem is referenced by:  bnj1501  29373  bnj1523  29377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-ex 1551  df-nf 1554  df-rex 2703
  Copyright terms: Public domain W3C validator