Theorem mpbiran3d 46030

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

Hypotheses

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

Assertion

Ref	Expression
mpbiran3d	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Proof of Theorem mpbiran3d

Step	Hyp	Ref	Expression
1		mpbiran3d.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))
2	1	simprbda 498	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3	2	ex 412	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
4		mpbiran3d.2	. . . . 5 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
5	4	ex 412	. . . 4 ⊢ (𝜑 → (𝜒 → 𝜃))
6	5	ancld 550	. . 3 ⊢ (𝜑 → (𝜒 → (𝜒 ∧ 𝜃)))
7	6, 1	sylibrd 258	. 2 ⊢ (𝜑 → (𝜒 → 𝜓))
8	3, 7	impbid 211	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ 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 206  df-an 396

This theorem is referenced by:  mpbiran4d  46031  functhinc  46214  thincsect  46226  thincinv  46228  grptcmon  46263  grptcepi  46264