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Metamath Proof Explorer

Theorem imbibi

Description: The antecedent of one side of a biconditional can be moved out of the biconditional to become the antecedent of the remaining biconditional. (Contributed by BJ, 1-Jan-2025) (Proof shortened by Wolf Lammen, 5-Jan-2025)

		Ref	Expression
	Assertion	imbibi	⊢ ( ( ( 𝜑 → 𝜓 ) ↔ 𝜒 ) → ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.4	⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) ↔ ( 𝜑 → 𝜓 ) )
2		imbi2	⊢ ( ( ( 𝜑 → 𝜓 ) ↔ 𝜒 ) → ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) ↔ ( 𝜑 → 𝜒 ) ) )
3	1 2	bitr3id	⊢ ( ( ( 𝜑 → 𝜓 ) ↔ 𝜒 ) → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) )
4	3	pm5.74rd	⊢ ( ( ( 𝜑 → 𝜓 ) ↔ 𝜒 ) → ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) )