Theorem e123

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	e123.1	$$⊢\left(\phi \to \psi \right)$$
	e123.2	$$⊢\left(\phi ,\chi \to \theta \right)$$
	e123.3	$$⊢\left(\phi ,\chi ,\tau \to \eta \right)$$
	e123.4	$$⊢\psi \to \left(\theta \to \left(\eta \to \zeta \right)\right)$$
Assertion	e123	$$⊢\left(\phi ,\chi ,\tau \to \zeta \right)$$

Proof

Step	Hyp	Ref	Expression
1		e123.1	$$⊢\left(\phi \to \psi \right)$$
2		e123.2	$$⊢\left(\phi ,\chi \to \theta \right)$$
3		e123.3	$$⊢\left(\phi ,\chi ,\tau \to \eta \right)$$
4		e123.4	$$⊢\psi \to \left(\theta \to \left(\eta \to \zeta \right)\right)$$
5	1	vd13	$$⊢\left(\phi ,\chi ,\tau \to \psi \right)$$
6	2	vd23	$$⊢\left(\phi ,\chi ,\tau \to \theta \right)$$
7	5 6 3 4	e333	$$⊢\left(\phi ,\chi ,\tau \to \zeta \right)$$