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Theorem bnj1019 29663
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 1842 . 2  |-  ( E. p ( ( th 
/\  ch  /\  et )  /\  ta )  <->  ( ( th  /\  ch  /\  et )  /\  E. p ta ) )
2 bnj258 29585 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  <->  ( ( th  /\  ch  /\  et )  /\  ta ) )
32exbii 1726 . 2  |-  ( E. p ( th  /\  ch  /\  ta  /\  et ) 
<->  E. p ( ( th  /\  ch  /\  et )  /\  ta )
)
4 df-bnj17 29564 . 2  |-  ( ( th  /\  ch  /\  et  /\  E. p ta )  <->  ( ( th 
/\  ch  /\  et )  /\  E. p ta ) )
51, 3, 43bitr4i 285 1  |-  ( E. p ( th  /\  ch  /\  ta  /\  et ) 
<->  ( th  /\  ch  /\  et  /\  E. p ta ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    /\ w3a 1007   E.wex 1671    /\ w-bnj17 29563
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-bnj17 29564
This theorem is referenced by:  bnj1018  29845  bnj1020  29846  bnj1021  29847
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