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Metamath Proof Explorer

Theorem el12

Description: Virtual deduction form of syl2an . (Contributed by Alan Sare, 23-Apr-2015) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	el12.1	$⊢ (φ \to ψ)$
	el12.2	$⊢ (τ \to χ)$
	el12.3	$⊢ (ψ \land χ) \to θ$
Assertion	el12	$⊢ ((φ, τ) \to θ)$

Proof

Step	Hyp	Ref	Expression
1		el12.1	$⊢ (φ \to ψ)$
2		el12.2	$⊢ (τ \to χ)$
3		el12.3	$⊢ (ψ \land χ) \to θ$
4	1	in1	$⊢ φ \to ψ$
5	2	in1	$⊢ τ \to χ$
6	4 5 3	syl2an	$⊢ (φ \land τ) \to θ$
7	6	dfvd2anir	$⊢ ((φ, τ) \to θ)$