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Metamath Proof Explorer

Theorem pm5.6

Description: Conjunction in antecedent versus disjunction in consequent. Theorem *5.6 of WhiteheadRussell p. 125. (Contributed by NM, 8-Jun-1994)

		Ref	Expression
	Assertion	pm5.6	⊢ ( ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝜒 ) ↔ ( 𝜑 → ( 𝜓 ∨ 𝜒 ) ) )

Step	Hyp	Ref	Expression
1		impexp	⊢ ( ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝜒 ) ↔ ( 𝜑 → ( ¬ 𝜓 → 𝜒 ) ) )
2		df-or	⊢ ( ( 𝜓 ∨ 𝜒 ) ↔ ( ¬ 𝜓 → 𝜒 ) )
3	2	imbi2i	⊢ ( ( 𝜑 → ( 𝜓 ∨ 𝜒 ) ) ↔ ( 𝜑 → ( ¬ 𝜓 → 𝜒 ) ) )
4	1 3	bitr4i	⊢ ( ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝜒 ) ↔ ( 𝜑 → ( 𝜓 ∨ 𝜒 ) ) )