Theorem el2122old

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	el2122old.1	$$⊢\left(\left(\phi ,\psi \right)\to \chi \right)$$
	el2122old.2	$$⊢\left(\psi \to \theta \right)$$
	el2122old.3	$$⊢\left(\psi \to \tau \right)$$
	el2122old.4	$$⊢\left(\chi \wedge \theta \wedge \tau \right)\to \eta$$
Assertion	el2122old	$$⊢\left(\left(\phi ,\psi \right)\to \eta \right)$$

Proof

Step	Hyp	Ref	Expression
1		el2122old.1	$$⊢\left(\left(\phi ,\psi \right)\to \chi \right)$$
2		el2122old.2	$$⊢\left(\psi \to \theta \right)$$
3		el2122old.3	$$⊢\left(\psi \to \tau \right)$$
4		el2122old.4	$$⊢\left(\chi \wedge \theta \wedge \tau \right)\to \eta$$
5	1	dfvd2ani	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
6	2	in1	$$⊢\psi \to \theta$$
7	3	in1	$$⊢\psi \to \tau$$
8	5 6 7 4	eel2122old	$$⊢\left(\phi \wedge \psi \right)\to \eta$$
9	8	dfvd2anir	$$⊢\left(\left(\phi ,\psi \right)\to \eta \right)$$