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Metamath Proof Explorer

Theorem pm5.74

Description: Distribution of implication over biconditional. Theorem *5.74 of WhiteheadRussell p. 126. (Contributed by NM, 1-Aug-1994) (Proof shortened by Wolf Lammen, 11-Apr-2013)

		Ref	Expression
	Assertion	pm5.74	⊢ ( ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) ↔ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		biimp	⊢ ( ( 𝜓 ↔ 𝜒 ) → ( 𝜓 → 𝜒 ) )
2	1	imim3i	⊢ ( ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )
3		biimpr	⊢ ( ( 𝜓 ↔ 𝜒 ) → ( 𝜒 → 𝜓 ) )
4	3	imim3i	⊢ ( ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) → ( ( 𝜑 → 𝜒 ) → ( 𝜑 → 𝜓 ) ) )
5	2 4	impbid	⊢ ( ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) )
6		biimp	⊢ ( ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )
7	6	pm2.86d	⊢ ( ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) → ( 𝜑 → ( 𝜓 → 𝜒 ) ) )
8		biimpr	⊢ ( ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜒 ) → ( 𝜑 → 𝜓 ) ) )
9	8	pm2.86d	⊢ ( ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) → ( 𝜑 → ( 𝜒 → 𝜓 ) ) )
10	7 9	impbidd	⊢ ( ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) → ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) )
11	5 10	impbii	⊢ ( ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) ↔ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) )