Theorem pm4.72

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)

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

Proof

Step	Hyp	Ref	Expression
1		olc	⊢ ( 𝜓 → ( 𝜑 ∨ 𝜓 ) )
2		pm2.621	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 ∨ 𝜓 ) → 𝜓 ) )
3	1 2	impbid2	⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜓 ↔ ( 𝜑 ∨ 𝜓 ) ) )
4		orc	⊢ ( 𝜑 → ( 𝜑 ∨ 𝜓 ) )
5		biimpr	⊢ ( ( 𝜓 ↔ ( 𝜑 ∨ 𝜓 ) ) → ( ( 𝜑 ∨ 𝜓 ) → 𝜓 ) )
6	4 5	syl5	⊢ ( ( 𝜓 ↔ ( 𝜑 ∨ 𝜓 ) ) → ( 𝜑 → 𝜓 ) )
7	3 6	impbii	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜓 ↔ ( 𝜑 ∨ 𝜓 ) ) )