Theorem anbiim 641

Description: Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.) (Proof shortened by Wolf Lammen, 7-May-2025.)

Hypotheses

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

Assertion

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

Proof of Theorem anbiim

Step	Hyp	Ref	Expression
1		anbiim.1	. . 3 ⊢ (𝜑 → (𝜒 → 𝜃))
2	1	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
3		anbiim.2	. . 3 ⊢ (𝜓 → (𝜃 → 𝜒))
4	3	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜃 → 𝜒))
5	2, 4	impbid 212	1 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 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:  sseq1  4009  sseq2  4010  gricsymb  47891  grlicsymb  47974