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Theorem bnj1521 33205
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 33149 . 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 33179 1  |-  ( ch 
->  E. x th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973   A.wal 1377   E.wex 1596    e. wcel 1767   E.wrex 2815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-ex 1597  df-nf 1600  df-rex 2820
This theorem is referenced by:  bnj1204  33364  bnj1311  33376  bnj1398  33386  bnj1408  33388  bnj1450  33402  bnj1312  33410  bnj1501  33419  bnj1523  33423
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