Theorem rb-bijust 1748

Description: Justification for rb-imdf 1749. (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 213	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)))
2		imor 853	. . . 4 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
3		imor 853	. . . . 5 ⊢ ((𝜓 → 𝜑) ↔ (¬ 𝜓 ∨ 𝜑))
4	3	notbii 320	. . . 4 ⊢ (¬ (𝜓 → 𝜑) ↔ ¬ (¬ 𝜓 ∨ 𝜑))
5	2, 4	imbi12i 350	. . 3 ⊢ (((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) ↔ ((¬ 𝜑 ∨ 𝜓) → ¬ (¬ 𝜓 ∨ 𝜑)))
6	5	notbii 320	. 2 ⊢ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) ↔ ¬ ((¬ 𝜑 ∨ 𝜓) → ¬ (¬ 𝜓 ∨ 𝜑)))
7		pm4.62 856	. . 3 ⊢ (((¬ 𝜑 ∨ 𝜓) → ¬ (¬ 𝜓 ∨ 𝜑)) ↔ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑)))
8	7	notbii 320	. 2 ⊢ (¬ ((¬ 𝜑 ∨ 𝜓) → ¬ (¬ 𝜓 ∨ 𝜑)) ↔ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑)))
9	1, 6, 8	3bitri 297	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ ¬ (¬ 𝜓 ∨ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 847

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 207  df-or 848

This theorem is referenced by:  rb-imdf  1749