Theorem e211

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	e211.1	$$⊢\left(\phi ,\psi \to \chi \right)$$
	e211.2	$$⊢\left(\phi \to \theta \right)$$
	e211.3	$$⊢\left(\phi \to \tau \right)$$
	e211.4	$$⊢\chi \to \left(\theta \to \left(\tau \to \eta \right)\right)$$
Assertion	e211	$$⊢\left(\phi ,\psi \to \eta \right)$$

Proof

Step	Hyp	Ref	Expression
1		e211.1	$$⊢\left(\phi ,\psi \to \chi \right)$$
2		e211.2	$$⊢\left(\phi \to \theta \right)$$
3		e211.3	$$⊢\left(\phi \to \tau \right)$$
4		e211.4	$$⊢\chi \to \left(\theta \to \left(\tau \to \eta \right)\right)$$
5	2	vd12	$$⊢\left(\phi ,\psi \to \theta \right)$$
6	3	vd12	$$⊢\left(\phi ,\psi \to \tau \right)$$
7	1 5 6 4	e222	$$⊢\left(\phi ,\psi \to \eta \right)$$