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Theorem bnj115 29067
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj115.1  |-  ( et  <->  A. n  e.  D  ( ta  ->  th )
)
Assertion
Ref Expression
bnj115  |-  ( et  <->  A. n ( ( n  e.  D  /\  ta )  ->  th ) )

Proof of Theorem bnj115
StepHypRef Expression
1 bnj115.1 . 2  |-  ( et  <->  A. n  e.  D  ( ta  ->  th )
)
2 df-ral 2561 . 2  |-  ( A. n  e.  D  ( ta  ->  th )  <->  A. n
( n  e.  D  ->  ( ta  ->  th )
) )
3 impexp 433 . . . 4  |-  ( ( ( n  e.  D  /\  ta )  ->  th )  <->  ( n  e.  D  -> 
( ta  ->  th )
) )
43bicomi 193 . . 3  |-  ( ( n  e.  D  -> 
( ta  ->  th )
)  <->  ( ( n  e.  D  /\  ta )  ->  th ) )
54albii 1556 . 2  |-  ( A. n ( n  e.  D  ->  ( ta  ->  th ) )  <->  A. n
( ( n  e.  D  /\  ta )  ->  th ) )
61, 2, 53bitri 262 1  |-  ( et  <->  A. n ( ( n  e.  D  /\  ta )  ->  th ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    e. wcel 1696   A.wral 2556
This theorem is referenced by:  bnj953  29287  bnj964  29291  bnj1090  29325  bnj1112  29329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547
This theorem depends on definitions:  df-bi 177  df-an 360  df-ral 2561
  Copyright terms: Public domain W3C validator