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Metamath Proof Explorer

Theorem e233

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 29-Feb-2012) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	e233.1	$⊢ (φ, ψ \to χ)$
	e233.2	$⊢ (φ, ψ, θ \to τ)$
	e233.3	$⊢ (φ, ψ, θ \to η)$
	e233.4	$⊢ χ \to (τ \to (η \to ζ))$
Assertion	e233	$⊢ (φ, ψ, θ \to ζ)$

Proof

Step	Hyp	Ref	Expression
1		e233.1	$⊢ (φ, ψ \to χ)$
2		e233.2	$⊢ (φ, ψ, θ \to τ)$
3		e233.3	$⊢ (φ, ψ, θ \to η)$
4		e233.4	$⊢ χ \to (τ \to (η \to ζ))$
5	1	dfvd2i	$⊢ φ \to (ψ \to χ)$
6	2	dfvd3i	$⊢ φ \to (ψ \to (θ \to τ))$
7	3	dfvd3i	$⊢ φ \to (ψ \to (θ \to η))$
8	5 6 7 4	ee233	$⊢ φ \to (ψ \to (θ \to ζ))$
9	8	dfvd3ir	$⊢ (φ, ψ, θ \to ζ)$