Theorem el123

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

	Ref	Expression
Hypotheses	el123.1	$$⊢\left(\phi \to \psi \right)$$
	el123.2	$$⊢\left(\chi \to \theta \right)$$
	el123.3	$$⊢\left(\tau \to \eta \right)$$
	el123.4	$$⊢\left(\psi \wedge \theta \wedge \eta \right)\to \zeta$$
Assertion	el123	$$⊢\left(\left(\phi ,\chi ,\tau \right)\to \zeta \right)$$

Proof

Step	Hyp	Ref	Expression
1		el123.1	$$⊢\left(\phi \to \psi \right)$$
2		el123.2	$$⊢\left(\chi \to \theta \right)$$
3		el123.3	$$⊢\left(\tau \to \eta \right)$$
4		el123.4	$$⊢\left(\psi \wedge \theta \wedge \eta \right)\to \zeta$$
5	1	in1	$$⊢\phi \to \psi$$
6	2	in1	$$⊢\chi \to \theta$$
7	3	in1	$$⊢\tau \to \eta$$
8	5 6 7 4	syl3an	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \zeta$$
9	8	dfvd3anir	$$⊢\left(\left(\phi ,\chi ,\tau \right)\to \zeta \right)$$