Theorem pm4.79

Description: Theorem *4.79 of WhiteheadRussell p. 121. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 27-Jun-2013)

	Ref	Expression
Assertion	pm4.79	⊢ ( ( ( 𝜓 → 𝜑 ) ∨ ( 𝜒 → 𝜑 ) ) ↔ ( ( 𝜓 ∧ 𝜒 ) → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜑 ) )
2		id	⊢ ( ( 𝜒 → 𝜑 ) → ( 𝜒 → 𝜑 ) )
3	1 2	jaoa	⊢ ( ( ( 𝜓 → 𝜑 ) ∨ ( 𝜒 → 𝜑 ) ) → ( ( 𝜓 ∧ 𝜒 ) → 𝜑 ) )
4		simplim	⊢ ( ¬ ( 𝜓 → 𝜑 ) → 𝜓 )
5		pm3.3	⊢ ( ( ( 𝜓 ∧ 𝜒 ) → 𝜑 ) → ( 𝜓 → ( 𝜒 → 𝜑 ) ) )
6	4 5	syl5	⊢ ( ( ( 𝜓 ∧ 𝜒 ) → 𝜑 ) → ( ¬ ( 𝜓 → 𝜑 ) → ( 𝜒 → 𝜑 ) ) )
7	6	orrd	⊢ ( ( ( 𝜓 ∧ 𝜒 ) → 𝜑 ) → ( ( 𝜓 → 𝜑 ) ∨ ( 𝜒 → 𝜑 ) ) )
8	3 7	impbii	⊢ ( ( ( 𝜓 → 𝜑 ) ∨ ( 𝜒 → 𝜑 ) ) ↔ ( ( 𝜓 ∧ 𝜒 ) → 𝜑 ) )