Theorem bi23imp13 42000

Description: 3imp 1109 with middle implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

Hypothesis

Ref	Expression
bi23imp13.1	⊢ (𝜑 → (𝜓 ↔ (𝜒 → 𝜃)))

Assertion

Ref	Expression
bi23imp13	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Proof of Theorem bi23imp13

Step	Hyp	Ref	Expression
1		bi23imp13.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜒 → 𝜃)))
2	1	biimpd 228	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
3	2	3imp 1109	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085

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  df-3an 1087

This theorem is referenced by:  bi12imp3  42003