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Theorem bnj1541 29289
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1541.1  |-  ( ph  <->  ( ps  /\  A  =/= 
B ) )
bnj1541.2  |-  -.  ph
Assertion
Ref Expression
bnj1541  |-  ( ps 
->  A  =  B
)

Proof of Theorem bnj1541
StepHypRef Expression
1 bnj1541.2 . . . 4  |-  -.  ph
2 bnj1541.1 . . . 4  |-  ( ph  <->  ( ps  /\  A  =/= 
B ) )
31, 2mtbi 291 . . 3  |-  -.  ( ps  /\  A  =/=  B
)
43imnani 414 . 2  |-  ( ps 
->  -.  A  =/=  B
)
5 nne 2607 . 2  |-  ( -.  A  =/=  B  <->  A  =  B )
64, 5sylib 190 1  |-  ( ps 
->  A  =  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    =/= wne 2601
This theorem is referenced by:  bnj1312  29489  bnj1523  29502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 179  df-an 362  df-ne 2603
  Copyright terms: Public domain W3C validator