Theorem rb-bijust 1514

Description: Justification for rb-imdf 1515. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rb-bijust	⊢ ((φ ↔ ψ) ↔ ¬ (¬ (¬ φ ∨ ψ) ∨ ¬ (¬ ψ ∨ φ)))

Proof of Theorem rb-bijust

Step	Hyp	Ref	Expression
1		dfbi1 184	. 2 ⊢ ((φ ↔ ψ) ↔ ¬ ((φ → ψ) → ¬ (ψ → φ)))
2		imor 401	. . . 4 ⊢ ((φ → ψ) ↔ (¬ φ ∨ ψ))
3		imor 401	. . . . 5 ⊢ ((ψ → φ) ↔ (¬ ψ ∨ φ))
4	3	notbii 287	. . . 4 ⊢ (¬ (ψ → φ) ↔ ¬ (¬ ψ ∨ φ))
5	2, 4	imbi12i 316	. . 3 ⊢ (((φ → ψ) → ¬ (ψ → φ)) ↔ ((¬ φ ∨ ψ) → ¬ (¬ ψ ∨ φ)))
6	5	notbii 287	. 2 ⊢ (¬ ((φ → ψ) → ¬ (ψ → φ)) ↔ ¬ ((¬ φ ∨ ψ) → ¬ (¬ ψ ∨ φ)))
7		pm4.62 408	. . 3 ⊢ (((¬ φ ∨ ψ) → ¬ (¬ ψ ∨ φ)) ↔ (¬ (¬ φ ∨ ψ) ∨ ¬ (¬ ψ ∨ φ)))
8	7	notbii 287	. 2 ⊢ (¬ ((¬ φ ∨ ψ) → ¬ (¬ ψ ∨ φ)) ↔ ¬ (¬ (¬ φ ∨ ψ) ∨ ¬ (¬ ψ ∨ φ)))
9	1, 6, 8	3bitri 262	1 ⊢ ((φ ↔ ψ) ↔ ¬ (¬ (¬ φ ∨ ψ) ∨ ¬ (¬ ψ ∨ φ)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357

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-or 359

This theorem is used by:  rb-imdf  1515