Theorem pm4.72 846

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 373	. . 3 ⊢ (ψ → (φ ∨ ψ))
2		pm2.621 397	. . 3 ⊢ ((φ → ψ) → ((φ ∨ ψ) → ψ))
3	1, 2	impbid2 195	. 2 ⊢ ((φ → ψ) → (ψ ↔ (φ ∨ ψ)))
4		orc 374	. . 3 ⊢ (φ → (φ ∨ ψ))
5		bi2 189	. . 3 ⊢ ((ψ ↔ (φ ∨ ψ)) → ((φ ∨ ψ) → ψ))
6	4, 5	syl5 28	. 2 ⊢ ((ψ ↔ (φ ∨ ψ)) → (φ → ψ))
7	3, 6	impbii 180	1 ⊢ ((φ → ψ) ↔ (ψ ↔ (φ ∨ ψ)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∨ wo 357

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359

This theorem is used by:  bigolden  901  ssequn1  3434  ssunsn2  3866