This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem exbir

Description: Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD . (Contributed by Alan Sare, 31-Dec-2011)

		Ref	Expression
	Assertion	exbir	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) ) → ( 𝜑 → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		biimpr	⊢ ( ( 𝜒 ↔ 𝜃 ) → ( 𝜃 → 𝜒 ) )
2	1	imim2i	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) ) → ( ( 𝜑 ∧ 𝜓 ) → ( 𝜃 → 𝜒 ) ) )
3	2	expd	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) ) → ( 𝜑 → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) )