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Metamath Proof Explorer

Theorem mpbiran4d

Description: Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 27-Sep-2024)

	Ref	Expression
Hypotheses	mpbiran3d.1	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) )
	mpbiran4d.2	⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜒 )
Assertion	mpbiran4d	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		mpbiran3d.1	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) )
2		mpbiran4d.2	⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜒 )
3	1	biancomd	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜃 ∧ 𝜒 ) ) )
4	3 2	mpbiran3d	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )