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Theorem bnj1019 29127
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 1858 . 2  |-  ( E. p ( ( th 
/\  ch  /\  et )  /\  ta )  <->  ( ( th  /\  ch  /\  et )  /\  E. p ta ) )
2 bnj258 29049 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  <->  ( ( th  /\  ch  /\  et )  /\  ta ) )
32exbii 1572 . 2  |-  ( E. p ( th  /\  ch  /\  ta  /\  et ) 
<->  E. p ( ( th  /\  ch  /\  et )  /\  ta )
)
4 df-bnj17 29028 . 2  |-  ( ( th  /\  ch  /\  et  /\  E. p ta )  <->  ( ( th 
/\  ch  /\  et )  /\  E. p ta ) )
51, 3, 43bitr4i 268 1  |-  ( E. p ( th  /\  ch  /\  ta  /\  et ) 
<->  ( th  /\  ch  /\  et  /\  E. p ta ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    /\ w-bnj17 29027
This theorem is referenced by:  bnj1018  29310  bnj1020  29311  bnj1021  29312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1532  df-nf 1535  df-bnj17 29028
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