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Metamath Proof Explorer

Theorem e12an

Description: Conjunction form of e12 (see syl6an ). (Contributed by Alan Sare, 11-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	e12an.1	$⊢ (φ \to ψ)$
	e12an.2	$⊢ (φ, χ \to θ)$
	e12an.3	$⊢ (ψ \land θ) \to τ$
Assertion	e12an	$⊢ (φ, χ \to τ)$

Proof

Step	Hyp	Ref	Expression
1		e12an.1	$⊢ (φ \to ψ)$
2		e12an.2	$⊢ (φ, χ \to θ)$
3		e12an.3	$⊢ (ψ \land θ) \to τ$
4	3	ex	$⊢ ψ \to (θ \to τ)$
5	1 2 4	e12	$⊢ (φ, χ \to τ)$