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Metamath Proof Explorer

Theorem e111

Description: A virtual deduction elimination rule (see syl3c ). (Contributed by Alan Sare, 14-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	e111.1	$⊢ (φ \to ψ)$
	e111.2	$⊢ (φ \to χ)$
	e111.3	$⊢ (φ \to θ)$
	e111.4	$⊢ ψ \to (χ \to (θ \to τ))$
Assertion	e111	$⊢ (φ \to τ)$

Proof

Step	Hyp	Ref	Expression
1		e111.1	$⊢ (φ \to ψ)$
2		e111.2	$⊢ (φ \to χ)$
3		e111.3	$⊢ (φ \to θ)$
4		e111.4	$⊢ ψ \to (χ \to (θ \to τ))$
5	3	in1	$⊢ φ \to θ$
6	1	in1	$⊢ φ \to ψ$
7	2	in1	$⊢ φ \to χ$
8	6 7 4	syl2im	$⊢ φ \to (φ \to (θ \to τ))$
9	8	pm2.43i	$⊢ φ \to (θ \to τ)$
10	5 9	syl5com	$⊢ φ \to (φ \to τ)$
11	10	pm2.43i	$⊢ φ \to τ$
12	11	dfvd1ir	$⊢ (φ \to τ)$