Theorem anbiim 653

Description: Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.) (Proof shortened by Wolf Lammen, 7-May-2025.) (Proof shortened by Garrett Katz, 15-Jun-2026.)

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		anbiim.2	. . 3 ⊢ (𝜓 → (𝜃 → 𝜒))
3	1, 2	impbid21d 214	. 2 ⊢ (𝜓 → (𝜑 → (𝜒 ↔ 𝜃)))
4	3	impcom 413	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:  sseq1  3956  sseq2  3957  ssdifsym  4220  wl-eujustlem1  38500  gricsymb  48989  grlicsymb  49081