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Metamath Proof Explorer

Theorem anbi1cd

Description: Introduce a proposition as left conjunct on the left-hand side and right conjunct on the right-hand side of an equivalence. Deduction form. (Contributed by Peter Mazsa, 22-May-2021)

		Ref	Expression
	Hypothesis	anbi1cd.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	Assertion	anbi1cd	⊢ ( 𝜑 → ( ( 𝜃 ∧ 𝜓 ) ↔ ( 𝜒 ∧ 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		anbi1cd.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	anbi2d	⊢ ( 𝜑 → ( ( 𝜃 ∧ 𝜓 ) ↔ ( 𝜃 ∧ 𝜒 ) ) )
3	2	biancomd	⊢ ( 𝜑 → ( ( 𝜃 ∧ 𝜓 ) ↔ ( 𝜒 ∧ 𝜃 ) ) )