Theorem bi123imp0

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

	Ref	Expression
Hypothesis	bi23imp0.1	⊢ ( 𝜑 ↔ ( 𝜓 ↔ ( 𝜒 ↔ 𝜃 ) ) )
Assertion	bi123imp0	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		bi23imp0.1	⊢ ( 𝜑 ↔ ( 𝜓 ↔ ( 𝜒 ↔ 𝜃 ) ) )
2		biimp	⊢ ( ( 𝜓 ↔ ( 𝜒 ↔ 𝜃 ) ) → ( 𝜓 → ( 𝜒 ↔ 𝜃 ) ) )
3		biimp	⊢ ( ( 𝜒 ↔ 𝜃 ) → ( 𝜒 → 𝜃 ) )
4	2 3	syl6	⊢ ( ( 𝜓 ↔ ( 𝜒 ↔ 𝜃 ) ) → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
5	1 4	sylbi	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
6	5	3imp	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )