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Theorem bnj1019 29543
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 1827 . 2  |-  ( E. p ( ( th 
/\  ch  /\  et )  /\  ta )  <->  ( ( th  /\  ch  /\  et )  /\  E. p ta ) )
2 bnj258 29465 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  <->  ( ( th  /\  ch  /\  et )  /\  ta ) )
32exbii 1712 . 2  |-  ( E. p ( th  /\  ch  /\  ta  /\  et ) 
<->  E. p ( ( th  /\  ch  /\  et )  /\  ta )
)
4 df-bnj17 29444 . 2  |-  ( ( th  /\  ch  /\  et  /\  E. p ta )  <->  ( ( th 
/\  ch  /\  et )  /\  E. p ta ) )
51, 3, 43bitr4i 280 1  |-  ( E. p ( th  /\  ch  /\  ta  /\  et ) 
<->  ( th  /\  ch  /\  et  /\  E. p ta ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    /\ w3a 982   E.wex 1657    /\ w-bnj17 29443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798
This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-ex 1658  df-bnj17 29444
This theorem is referenced by:  bnj1018  29725  bnj1020  29726  bnj1021  29727
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