Theorem pm5.32

Description: Distribution of implication over biconditional. Theorem *5.32 of WhiteheadRussell p. 125. (Contributed by NM, 1-Aug-1994)

	Ref	Expression
Assertion	pm5.32	⊢ ( ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		notbi	⊢ ( ( 𝜓 ↔ 𝜒 ) ↔ ( ¬ 𝜓 ↔ ¬ 𝜒 ) )
2	1	imbi2i	⊢ ( ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) ↔ ( 𝜑 → ( ¬ 𝜓 ↔ ¬ 𝜒 ) ) )
3		pm5.74	⊢ ( ( 𝜑 → ( ¬ 𝜓 ↔ ¬ 𝜒 ) ) ↔ ( ( 𝜑 → ¬ 𝜓 ) ↔ ( 𝜑 → ¬ 𝜒 ) ) )
4		notbi	⊢ ( ( ( 𝜑 → ¬ 𝜓 ) ↔ ( 𝜑 → ¬ 𝜒 ) ) ↔ ( ¬ ( 𝜑 → ¬ 𝜓 ) ↔ ¬ ( 𝜑 → ¬ 𝜒 ) ) )
5	2 3 4	3bitri	⊢ ( ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) ↔ ( ¬ ( 𝜑 → ¬ 𝜓 ) ↔ ¬ ( 𝜑 → ¬ 𝜒 ) ) )
6		df-an	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ¬ ( 𝜑 → ¬ 𝜓 ) )
7		df-an	⊢ ( ( 𝜑 ∧ 𝜒 ) ↔ ¬ ( 𝜑 → ¬ 𝜒 ) )
8	6 7	bibi12i	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜒 ) ) ↔ ( ¬ ( 𝜑 → ¬ 𝜓 ) ↔ ¬ ( 𝜑 → ¬ 𝜒 ) ) )
9	5 8	bitr4i	⊢ ( ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜒 ) ) )