Theorem e323

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 17-Apr-2012) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	e323.1	$$⊢\left(\phi ,\psi ,\chi \to \theta \right)$$
	e323.2	$$⊢\left(\phi ,\psi \to \tau \right)$$
	e323.3	$$⊢\left(\phi ,\psi ,\chi \to \eta \right)$$
	e323.4	$$⊢\theta \to \left(\tau \to \left(\eta \to \zeta \right)\right)$$
Assertion	e323	$$⊢\left(\phi ,\psi ,\chi \to \zeta \right)$$

Proof

Step	Hyp	Ref	Expression
1		e323.1	$$⊢\left(\phi ,\psi ,\chi \to \theta \right)$$
2		e323.2	$$⊢\left(\phi ,\psi \to \tau \right)$$
3		e323.3	$$⊢\left(\phi ,\psi ,\chi \to \eta \right)$$
4		e323.4	$$⊢\theta \to \left(\tau \to \left(\eta \to \zeta \right)\right)$$
5	1	dfvd3i	$$⊢\phi \to \left(\psi \to \left(\chi \to \theta \right)\right)$$
6	2	dfvd2i	$$⊢\phi \to \left(\psi \to \tau \right)$$
7	3	dfvd3i	$$⊢\phi \to \left(\psi \to \left(\chi \to \eta \right)\right)$$
8	5 6 7 4	ee323	$$⊢\phi \to \left(\psi \to \left(\chi \to \zeta \right)\right)$$
9	8	dfvd3ir	$$⊢\left(\phi ,\psi ,\chi \to \zeta \right)$$