Theorem anbiimOLD 654

Description: Obsolete version of anbiim 653 as of 15-Jun-2026. Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.) (Proof shortened by Wolf Lammen, 7-May-2025.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
anbiim.1	⊢ (𝜑 → (𝜒 → 𝜃))
anbiim.2	⊢ (𝜓 → (𝜃 → 𝜒))

Assertion

Ref	Expression
anbiimOLD	⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))

Proof of Theorem anbiimOLD

Step	Hyp	Ref	Expression
1		anbiim.1	. . 3 ⊢ (𝜑 → (𝜒 → 𝜃))
2	1	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
3		anbiim.2	. . 3 ⊢ (𝜓 → (𝜃 → 𝜒))
4	3	adantl 487	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜃 → 𝜒))
5	2, 4	impbid 215	1 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by: (None)