Theorem mpbiran4d 48676

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

Hypotheses

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

Assertion

Ref	Expression
mpbiran4d	⊢ (𝜑 → (𝜓 ↔ 𝜃))

Proof of Theorem mpbiran4d

Step	Hyp	Ref	Expression
1		mpbiran3d.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))
2	1	biancomd 463	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜃 ∧ 𝜒)))
3		mpbiran4d.2	. 2 ⊢ ((𝜑 ∧ 𝜃) → 𝜒)
4	2, 3	mpbiran3d 48675	1 ⊢ (𝜑 → (𝜓 ↔ 𝜃))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by: (None)