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Theorem aifftbifffaibifff 38229
Description: Given a is equivalent to T., Given b is equivalent to F., there exists a proof for that a iff b is false. (Contributed by Jarvin Udandy, 7-Sep-2020.)
Hypotheses
Ref Expression
aifftbifffaibifff.1  |-  ( ph  <-> T.  )
aifftbifffaibifff.2  |-  ( ps  <-> F.  )
Assertion
Ref Expression
aifftbifffaibifff  |-  ( (
ph 
<->  ps )  <-> F.  )

Proof of Theorem aifftbifffaibifff
StepHypRef Expression
1 aifftbifffaibifff.1 . . . . 5  |-  ( ph  <-> T.  )
21aistia 38203 . . . 4  |-  ph
3 aifftbifffaibifff.2 . . . . 5  |-  ( ps  <-> F.  )
43aisfina 38204 . . . 4  |-  -.  ps
52, 4abnotbtaxb 38222 . . 3  |-  ( ph  \/_ 
ps )
65axorbtnotaiffb 38209 . 2  |-  -.  ( ph 
<->  ps )
7 nbfal 1449 . . 3  |-  ( -.  ( ph  <->  ps )  <->  ( ( ph  <->  ps )  <-> F.  ) )
87biimpi 198 . 2  |-  ( -.  ( ph  <->  ps )  ->  ( ( ph  <->  ps )  <-> F.  ) )
96, 8ax-mp 5 1  |-  ( (
ph 
<->  ps )  <-> F.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188   T. wtru 1439   F. wfal 1443
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 189  df-an 373  df-xor 1402  df-tru 1441  df-fal 1444
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator