Metamath Proof Explorer

Theorem ee222

Description: e222 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ee222.1	$⊢ φ \to (ψ \to χ)$
	ee222.2	$⊢ φ \to (ψ \to θ)$
	ee222.3	$⊢ φ \to (ψ \to τ)$
	ee222.4	$⊢ χ \to (θ \to (τ \to η))$
Assertion	ee222	$⊢ φ \to (ψ \to η)$

Proof

Step	Hyp	Ref	Expression
1		ee222.1	$⊢ φ \to (ψ \to χ)$
2		ee222.2	$⊢ φ \to (ψ \to θ)$
3		ee222.3	$⊢ φ \to (ψ \to τ)$
4		ee222.4	$⊢ χ \to (θ \to (τ \to η))$
5	1	imp	$⊢ (φ \land ψ) \to χ$
6	2	imp	$⊢ (φ \land ψ) \to θ$
7	3	imp	$⊢ (φ \land ψ) \to τ$
8	5 6 7 4	syl3c	$⊢ (φ \land ψ) \to η$
9	8	ex	$⊢ φ \to (ψ \to η)$