Theorem e112

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

	Ref	Expression
Hypotheses	e112.1	$$⊢\left(\phi \to \psi \right)$$
	e112.2	$$⊢\left(\phi \to \chi \right)$$
	e112.3	$$⊢\left(\phi ,\theta \to \tau \right)$$
	e112.4	$$⊢\psi \to \left(\chi \to \left(\tau \to \eta \right)\right)$$
Assertion	e112	$$⊢\left(\phi ,\theta \to \eta \right)$$

Proof

Step	Hyp	Ref	Expression
1		e112.1	$$⊢\left(\phi \to \psi \right)$$
2		e112.2	$$⊢\left(\phi \to \chi \right)$$
3		e112.3	$$⊢\left(\phi ,\theta \to \tau \right)$$
4		e112.4	$$⊢\psi \to \left(\chi \to \left(\tau \to \eta \right)\right)$$
5	1	vd12	$$⊢\left(\phi ,\theta \to \psi \right)$$
6	2	vd12	$$⊢\left(\phi ,\theta \to \chi \right)$$
7	5 6 3 4	e222	$$⊢\left(\phi ,\theta \to \eta \right)$$