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Theorem bnj1541 34300
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 296 . . 3  |-  -.  ( ps  /\  A  =/=  B
)
43imnani 421 . 2  |-  ( ps 
->  -.  A  =/=  B
)
5 nne 2597 . 2  |-  ( -.  A  =/=  B  <->  A  =  B )
64, 5sylib 196 1  |-  ( ps 
->  A  =  B
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    =/= wne 2591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 369  df-ne 2593
This theorem is referenced by:  bnj1312  34500  bnj1523  34513
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