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Metamath Proof Explorer

Theorem e1111

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 6-Mar-2012) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	e1111.1	$⊢ (φ \to ψ)$
	e1111.2	$⊢ (φ \to χ)$
	e1111.3	$⊢ (φ \to θ)$
	e1111.4	$⊢ (φ \to τ)$
	e1111.5	$⊢ ψ \to (χ \to (θ \to (τ \to η)))$
Assertion	e1111	$⊢ (φ \to η)$

Proof

Step	Hyp	Ref	Expression
1		e1111.1	$⊢ (φ \to ψ)$
2		e1111.2	$⊢ (φ \to χ)$
3		e1111.3	$⊢ (φ \to θ)$
4		e1111.4	$⊢ (φ \to τ)$
5		e1111.5	$⊢ ψ \to (χ \to (θ \to (τ \to η)))$
6	1	in1	$⊢ φ \to ψ$
7	2	in1	$⊢ φ \to χ$
8	3	in1	$⊢ φ \to θ$
9	4	in1	$⊢ φ \to τ$
10	6 7 8 9 5	ee1111	$⊢ φ \to η$
11	10	dfvd1ir	$⊢ (φ \to η)$