Theorem pm4.72 839

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 723	. . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
2		pm2.621 759	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 ∨ 𝜓) → 𝜓))
3	1, 2	impbid2 143	. 2 ⊢ ((𝜑 → 𝜓) → (𝜓 ↔ (𝜑 ∨ 𝜓)))
4		orc 724	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
5		biimpr 130	. . 3 ⊢ ((𝜓 ↔ (𝜑 ∨ 𝜓)) → ((𝜑 ∨ 𝜓) → 𝜓))
6	4, 5	syl5 32	. 2 ⊢ ((𝜓 ↔ (𝜑 ∨ 𝜓)) → (𝜑 → 𝜓))
7	3, 6	impbii 126	1 ⊢ ((𝜑 → 𝜓) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   ∨ wo 720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721

This proof depends on definitions:  df-bi 117

This theorem is used by:  bigolden  968  ssequn1  3399  vtxd0nedgbfi  16706