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Theorem bnj1019 29212
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1019  |-  ( E. p ( th  /\  ch  /\  ta  /\  et ) 
<->  ( th  /\  ch  /\  et  /\  E. p ta ) )
Distinct variable groups:    ch, p    et, p    th, p
Allowed substitution hint:    ta( p)

Proof of Theorem bnj1019
StepHypRef Expression
1 19.42v 1929 . 2  |-  ( E. p ( ( th 
/\  ch  /\  et )  /\  ta )  <->  ( ( th  /\  ch  /\  et )  /\  E. p ta ) )
2 bnj258 29134 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  <->  ( ( th  /\  ch  /\  et )  /\  ta ) )
32exbii 1593 . 2  |-  ( E. p ( th  /\  ch  /\  ta  /\  et ) 
<->  E. p ( ( th  /\  ch  /\  et )  /\  ta )
)
4 df-bnj17 29113 . 2  |-  ( ( th  /\  ch  /\  et  /\  E. p ta )  <->  ( ( th 
/\  ch  /\  et )  /\  E. p ta ) )
51, 3, 43bitr4i 270 1  |-  ( E. p ( th  /\  ch  /\  ta  /\  et ) 
<->  ( th  /\  ch  /\  et  /\  E. p ta ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    /\ w-bnj17 29112
This theorem is referenced by:  bnj1018  29395  bnj1020  29396  bnj1021  29397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 939  df-ex 1552  df-nf 1555  df-bnj17 29113
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