Theorem bi33imp12

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

	Ref	Expression
Hypothesis	bi33imp12.1	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ↔ 𝜃 ) ) )
Assertion	bi33imp12	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		bi33imp12.1	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ↔ 𝜃 ) ) )
2		biimp	⊢ ( ( 𝜒 ↔ 𝜃 ) → ( 𝜒 → 𝜃 ) )
3	1 2	syl6	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
4	3	3imp	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )