Theorem e233

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 29-Feb-2012) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	e233.1	$$⊢\left(\phi ,\psi \to \chi \right)$$
	e233.2	$$⊢\left(\phi ,\psi ,\theta \to \tau \right)$$
	e233.3	$$⊢\left(\phi ,\psi ,\theta \to \eta \right)$$
	e233.4	$$⊢\chi \to \left(\tau \to \left(\eta \to \zeta \right)\right)$$
Assertion	e233	$$⊢\left(\phi ,\psi ,\theta \to \zeta \right)$$

Proof

Step	Hyp	Ref	Expression
1		e233.1	$$⊢\left(\phi ,\psi \to \chi \right)$$
2		e233.2	$$⊢\left(\phi ,\psi ,\theta \to \tau \right)$$
3		e233.3	$$⊢\left(\phi ,\psi ,\theta \to \eta \right)$$
4		e233.4	$$⊢\chi \to \left(\tau \to \left(\eta \to \zeta \right)\right)$$
5	1	dfvd2i	$$⊢\phi \to \left(\psi \to \chi \right)$$
6	2	dfvd3i	$$⊢\phi \to \left(\psi \to \left(\theta \to \tau \right)\right)$$
7	3	dfvd3i	$$⊢\phi \to \left(\psi \to \left(\theta \to \eta \right)\right)$$
8	5 6 7 4	ee233	$$⊢\phi \to \left(\psi \to \left(\theta \to \zeta \right)\right)$$
9	8	dfvd3ir	$$⊢\left(\phi ,\psi ,\theta \to \zeta \right)$$