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Theorem tbw-bijust 1463
 Description: Justification for tbw-negdf 1464. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
tbw-bijust ((φψ) ↔ (((φψ) → ((ψφ) → ⊥ )) → ⊥ ))

Proof of Theorem tbw-bijust
StepHypRef Expression
1 dfbi1 184 . 2 ((φψ) ↔ ¬ ((φψ) → ¬ (ψφ)))
2 pm2.21 100 . . . . 5 (¬ (ψφ) → ((ψφ) → ⊥ ))
32imim2i 13 . . . 4 (((φψ) → ¬ (ψφ)) → ((φψ) → ((ψφ) → ⊥ )))
4 id 19 . . . . . 6 (¬ (ψφ) → ¬ (ψφ))
5 falim 1328 . . . . . 6 ( ⊥ → ¬ (ψφ))
64, 5ja 153 . . . . 5 (((ψφ) → ⊥ ) → ¬ (ψφ))
76imim2i 13 . . . 4 (((φψ) → ((ψφ) → ⊥ )) → ((φψ) → ¬ (ψφ)))
83, 7impbii 180 . . 3 (((φψ) → ¬ (ψφ)) ↔ ((φψ) → ((ψφ) → ⊥ )))
98notbii 287 . 2 (¬ ((φψ) → ¬ (ψφ)) ↔ ¬ ((φψ) → ((ψφ) → ⊥ )))
10 pm2.21 100 . . 3 (¬ ((φψ) → ((ψφ) → ⊥ )) → (((φψ) → ((ψφ) → ⊥ )) → ⊥ ))
11 ax-1 5 . . . . 5 (¬ ((φψ) → ((ψφ) → ⊥ )) → ((((φψ) → ((ψφ) → ⊥ )) → ⊥ ) → ¬ ((φψ) → ((ψφ) → ⊥ ))))
12 falim 1328 . . . . 5 ( ⊥ → ((((φψ) → ((ψφ) → ⊥ )) → ⊥ ) → ¬ ((φψ) → ((ψφ) → ⊥ ))))
1311, 12ja 153 . . . 4 ((((φψ) → ((ψφ) → ⊥ )) → ⊥ ) → ((((φψ) → ((ψφ) → ⊥ )) → ⊥ ) → ¬ ((φψ) → ((ψφ) → ⊥ ))))
1413pm2.43i 43 . . 3 ((((φψ) → ((ψφ) → ⊥ )) → ⊥ ) → ¬ ((φψ) → ((ψφ) → ⊥ )))
1510, 14impbii 180 . 2 (¬ ((φψ) → ((ψφ) → ⊥ )) ↔ (((φψ) → ((ψφ) → ⊥ )) → ⊥ ))
161, 9, 153bitri 262 1 ((φψ) ↔ (((φψ) → ((ψφ) → ⊥ )) → ⊥ ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ⊥ wfal 1317 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 177  df-tru 1319  df-fal 1320 This theorem is referenced by:  tbw-negdf  1464
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