Theorem bi123imp0 42005

Description: Similar to 3imp 1109 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

Hypothesis

Ref	Expression
bi23imp0.1	⊢ (𝜑 ↔ (𝜓 ↔ (𝜒 ↔ 𝜃)))

Assertion

Ref	Expression
bi123imp0	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Proof of Theorem bi123imp0

Step	Hyp	Ref	Expression
1		bi23imp0.1	. . 3 ⊢ (𝜑 ↔ (𝜓 ↔ (𝜒 ↔ 𝜃)))
2		biimp 214	. . . 4 ⊢ ((𝜓 ↔ (𝜒 ↔ 𝜃)) → (𝜓 → (𝜒 ↔ 𝜃)))
3		biimp 214	. . . 4 ⊢ ((𝜒 ↔ 𝜃) → (𝜒 → 𝜃))
4	2, 3	syl6 35	. . 3 ⊢ ((𝜓 ↔ (𝜒 ↔ 𝜃)) → (𝜓 → (𝜒 → 𝜃)))
5	1, 4	sylbi 216	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
6	5	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: (None)