Theorem pm4.72 920

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 399	. . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
2		pm2.621 424	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 ∨ 𝜓) → 𝜓))
3	1, 2	impbid2 216	. 2 ⊢ ((𝜑 → 𝜓) → (𝜓 ↔ (𝜑 ∨ 𝜓)))
4		orc 400	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
5		biimpr 210	. . 3 ⊢ ((𝜓 ↔ (𝜑 ∨ 𝜓)) → ((𝜑 ∨ 𝜓) → 𝜓))
6	4, 5	syl5 34	. 2 ⊢ ((𝜓 ↔ (𝜑 ∨ 𝜓)) → (𝜑 → 𝜓))
7	3, 6	impbii 199	1 ⊢ ((𝜑 → 𝜓) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383

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 197  df-or 385

This theorem is referenced by:  bigolden  976  cadan  1548  ssequn1  3783  ssunsn2  4359  vtxd0nedgb  26384  bj-consensusALT  32563  elpaddn0  35086