Theorem andiff 40159

Description: Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.)

Hypotheses

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

Assertion

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

Proof of Theorem andiff

Step	Hyp	Ref	Expression
1		andiff.1	. . 3 ⊢ (𝜑 → (𝜒 → 𝜃))
2		andiff.2	. . 3 ⊢ (𝜓 → (𝜃 → 𝜒))
3	1, 2	anim12i 613	. 2 ⊢ ((𝜑 ∧ 𝜓) → ((𝜒 → 𝜃) ∧ (𝜃 → 𝜒)))
4		dfbi2 475	. 2 ⊢ ((𝜒 ↔ 𝜃) ↔ ((𝜒 → 𝜃) ∧ (𝜃 → 𝜒)))
5	3, 4	sylibr 233	1 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396

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 397

This theorem is referenced by: (None)