Theorem e31an

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	e31an.1	$$⊢\left(\phi ,\psi ,\chi \to \theta \right)$$
	e31an.2	$$⊢\left(\phi \to \tau \right)$$
	e31an.3	$$⊢\left(\theta \wedge \tau \right)\to \eta$$
Assertion	e31an	$$⊢\left(\phi ,\psi ,\chi \to \eta \right)$$

Proof

Step	Hyp	Ref	Expression
1		e31an.1	$$⊢\left(\phi ,\psi ,\chi \to \theta \right)$$
2		e31an.2	$$⊢\left(\phi \to \tau \right)$$
3		e31an.3	$$⊢\left(\theta \wedge \tau \right)\to \eta$$
4	3	ex	$$⊢\theta \to \left(\tau \to \eta \right)$$
5	1 2 4	e31	$$⊢\left(\phi ,\psi ,\chi \to \eta \right)$$