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Theorem bnj1209 29680
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 29678 . . . 4  |-  ( ch 
->  E. x ( x  e.  B  /\  ph ) )
32ancli 560 . . 3  |-  ( ch 
->  ( ch  /\  E. x ( x  e.  B  /\  ph )
) )
4 19.42v 1842 . . 3  |-  ( E. x ( ch  /\  ( x  e.  B  /\  ph ) )  <->  ( ch  /\ 
E. x ( x  e.  B  /\  ph ) ) )
53, 4sylibr 217 . 2  |-  ( ch 
->  E. x ( ch 
/\  ( x  e.  B  /\  ph )
) )
6 bnj1209.2 . . 3  |-  ( th  <->  ( ch  /\  x  e.  B  /\  ph )
)
7 3anass 1011 . . 3  |-  ( ( ch  /\  x  e.  B  /\  ph )  <->  ( ch  /\  ( x  e.  B  /\  ph ) ) )
86, 7bitri 257 . 2  |-  ( th  <->  ( ch  /\  ( x  e.  B  /\  ph ) ) )
95, 8bnj1198 29679 1  |-  ( ch 
->  E. x th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007   E.wex 1671    e. wcel 1904   E.wrex 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813
This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-ex 1672  df-rex 2762
This theorem is referenced by:  bnj1501  29948  bnj1523  29952
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