Metamath Proof Explorer

Theorem e222

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Hypotheses	e222.1	$⊢ (φ, ψ \to χ)$
		e222.2	$⊢ (φ, ψ \to θ)$
		e222.3	$⊢ (φ, ψ \to τ)$
		e222.4	$⊢ χ \to (θ \to (τ \to η))$
	Assertion	e222	$⊢ (φ, ψ \to η)$

Proof

Step	Hyp	Ref	Expression
1		e222.1	$⊢ (φ, ψ \to χ)$
2		e222.2	$⊢ (φ, ψ \to θ)$
3		e222.3	$⊢ (φ, ψ \to τ)$
4		e222.4	$⊢ χ \to (θ \to (τ \to η))$
5	3	dfvd2i	$⊢ φ \to (ψ \to τ)$
6	5	imp	$⊢ (φ \land ψ) \to τ$
7	1	dfvd2i	$⊢ φ \to (ψ \to χ)$
8	7	imp	$⊢ (φ \land ψ) \to χ$
9	2	dfvd2i	$⊢ φ \to (ψ \to θ)$
10	9	imp	$⊢ (φ \land ψ) \to θ$
11	8 10 4	syl2im	$⊢ (φ \land ψ) \to ((φ \land ψ) \to (τ \to η))$
12	11	pm2.43i	$⊢ (φ \land ψ) \to (τ \to η)$
13	6 12	syl5com	$⊢ (φ \land ψ) \to ((φ \land ψ) \to η)$
14	13	pm2.43i	$⊢ (φ \land ψ) \to η$
15	14	ex	$⊢ φ \to (ψ \to η)$
16	15	dfvd2ir	$⊢ (φ, ψ \to η)$