Theorem pm4.72 736

Description: Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Jan-2013.)

Assertion

Ref	Expression
pm4.72	⊢ ((𝜑 → 𝜓) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓)))

Proof of Theorem pm4.72

Step	Hyp	Ref	Expression
1		olc 632	. . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
2		pm2.621 666	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 ∨ 𝜓) → 𝜓))
3	1, 2	impbid2 131	. 2 ⊢ ((𝜑 → 𝜓) → (𝜓 ↔ (𝜑 ∨ 𝜓)))
4		orc 633	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
5		bi2 121	. . 3 ⊢ ((𝜓 ↔ (𝜑 ∨ 𝜓)) → ((𝜑 ∨ 𝜓) → 𝜓))
6	4, 5	syl5 28	. 2 ⊢ ((𝜓 ↔ (𝜑 ∨ 𝜓)) → (𝜑 → 𝜓))
7	3, 6	impbii 117	1 ⊢ ((𝜑 → 𝜓) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 98   ∨ wo 629

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  bigolden  862  ssequn1  3113