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	⊢ ((φ ↔ ψ) ↔ (((φ → ψ) → ((ψ → φ) → ⊥ )) → ⊥ ))

Proof of Theorem tbw-bijust

Step	Hyp	Ref	Expression
1		dfbi1 184	. 2 ⊢ ((φ ↔ ψ) ↔ ¬ ((φ → ψ) → ¬ (ψ → φ)))
2		pm2.21 100	. . . . 5 ⊢ (¬ (ψ → φ) → ((ψ → φ) → ⊥ ))
3	2	imim2i 13	. . . 4 ⊢ (((φ → ψ) → ¬ (ψ → φ)) → ((φ → ψ) → ((ψ → φ) → ⊥ )))
4		id 19	. . . . . 6 ⊢ (¬ (ψ → φ) → ¬ (ψ → φ))
5		falim 1328	. . . . . 6 ⊢ ( ⊥ → ¬ (ψ → φ))
6	4, 5	ja 153	. . . . 5 ⊢ (((ψ → φ) → ⊥ ) → ¬ (ψ → φ))
7	6	imim2i 13	. . . 4 ⊢ (((φ → ψ) → ((ψ → φ) → ⊥ )) → ((φ → ψ) → ¬ (ψ → φ)))
8	3, 7	impbii 180	. . 3 ⊢ (((φ → ψ) → ¬ (ψ → φ)) ↔ ((φ → ψ) → ((ψ → φ) → ⊥ )))
9	8	notbii 287	. 2 ⊢ (¬ ((φ → ψ) → ¬ (ψ → φ)) ↔ ¬ ((φ → ψ) → ((ψ → φ) → ⊥ )))
10		pm2.21 100	. . 3 ⊢ (¬ ((φ → ψ) → ((ψ → φ) → ⊥ )) → (((φ → ψ) → ((ψ → φ) → ⊥ )) → ⊥ ))
11		ax-1 6	. . . . 5 ⊢ (¬ ((φ → ψ) → ((ψ → φ) → ⊥ )) → ((((φ → ψ) → ((ψ → φ) → ⊥ )) → ⊥ ) → ¬ ((φ → ψ) → ((ψ → φ) → ⊥ ))))
12		falim 1328	. . . . 5 ⊢ ( ⊥ → ((((φ → ψ) → ((ψ → φ) → ⊥ )) → ⊥ ) → ¬ ((φ → ψ) → ((ψ → φ) → ⊥ ))))
13	11, 12	ja 153	. . . 4 ⊢ ((((φ → ψ) → ((ψ → φ) → ⊥ )) → ⊥ ) → ((((φ → ψ) → ((ψ → φ) → ⊥ )) → ⊥ ) → ¬ ((φ → ψ) → ((ψ → φ) → ⊥ ))))
14	13	pm2.43i 43	. . 3 ⊢ ((((φ → ψ) → ((ψ → φ) → ⊥ )) → ⊥ ) → ¬ ((φ → ψ) → ((ψ → φ) → ⊥ )))
15	10, 14	impbii 180	. 2 ⊢ (¬ ((φ → ψ) → ((ψ → φ) → ⊥ )) ↔ (((φ → ψ) → ((ψ → φ) → ⊥ )) → ⊥ ))
16	1, 9, 15	3bitri 262	1 ⊢ ((φ ↔ ψ) ↔ (((φ → ψ) → ((ψ → φ) → ⊥ )) → ⊥ ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ⊥ wfal 1317

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-tru 1319  df-fal 1320

This theorem is used by:  tbw-negdf  1464