Theorem mpbiran3d

Description: Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 24-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		mpbiran3d.1	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) )
2		mpbiran3d.2	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
3	1	simprbda	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
4	3	ex	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
5	2	ex	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
6	5	ancld	⊢ ( 𝜑 → ( 𝜒 → ( 𝜒 ∧ 𝜃 ) ) )
7	6 1	sylibrd	⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
8	4 7	impbid	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )