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Metamath Proof Explorer

Theorem e123

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

	Ref	Expression
Hypotheses	e123.1	$⊢ (φ \to ψ)$
	e123.2	$⊢ (φ, χ \to θ)$
	e123.3	$⊢ (φ, χ, τ \to η)$
	e123.4	$⊢ ψ \to (θ \to (η \to ζ))$
Assertion	e123	$⊢ (φ, χ, τ \to ζ)$

Proof

Step	Hyp	Ref	Expression
1		e123.1	$⊢ (φ \to ψ)$
2		e123.2	$⊢ (φ, χ \to θ)$
3		e123.3	$⊢ (φ, χ, τ \to η)$
4		e123.4	$⊢ ψ \to (θ \to (η \to ζ))$
5	1	vd13	$⊢ (φ, χ, τ \to ψ)$
6	2	vd23	$⊢ (φ, χ, τ \to θ)$
7	5 6 3 4	e333	$⊢ (φ, χ, τ \to ζ)$