Theorem oibabs 709

Description: Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)

Assertion

Ref	Expression
oibabs	⊢ (((𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)) ↔ (𝜑 ↔ 𝜓))

Proof of Theorem oibabs

Step	Hyp	Ref	Expression
1		pm2.67-2 708	. . . 4 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)) → (𝜑 → (𝜑 ↔ 𝜓)))
2	1	ibd 177	. . 3 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)) → (𝜑 → 𝜓))
3		olc 706	. . . . 5 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
4	3	imim1i 60	. . . 4 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)) → (𝜓 → (𝜑 ↔ 𝜓)))
5		ibibr 245	. . . 4 ⊢ ((𝜓 → 𝜑) ↔ (𝜓 → (𝜑 ↔ 𝜓)))
6	4, 5	sylibr 133	. . 3 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)) → (𝜓 → 𝜑))
7	2, 6	impbid 128	. 2 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)) → (𝜑 ↔ 𝜓))
8		ax-1 6	. 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)))
9	7, 8	impbii 125	1 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)) ↔ (𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)