Theorem bi123imp0 45423

Description: Similar to 3imp 1128 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 218	. . . 4 ⊢ ((𝜓 ↔ (𝜒 ↔ 𝜃)) → (𝜓 → (𝜒 ↔ 𝜃)))
3		biimp 218	. . . 4 ⊢ ((𝜒 ↔ 𝜃) → (𝜒 → 𝜃))
4	2, 3	syl6 36	. . 3 ⊢ ((𝜓 ↔ (𝜒 ↔ 𝜃)) → (𝜓 → (𝜒 → 𝜃)))
5	1, 4	sylbi 220	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
6	5	3imp 1128	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103

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  df-3an 1105

This theorem is used by: (None)