Theorem e121

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

	Ref	Expression
Hypotheses	e121.1	$$⊢\left(\phi \to \psi \right)$$
	e121.2	$$⊢\left(\phi ,\chi \to \theta \right)$$
	e121.3	$$⊢\left(\phi \to \tau \right)$$
	e121.4	$$⊢\psi \to \left(\theta \to \left(\tau \to \eta \right)\right)$$
Assertion	e121	$$⊢\left(\phi ,\chi \to \eta \right)$$

Proof

Step	Hyp	Ref	Expression
1		e121.1	$$⊢\left(\phi \to \psi \right)$$
2		e121.2	$$⊢\left(\phi ,\chi \to \theta \right)$$
3		e121.3	$$⊢\left(\phi \to \tau \right)$$
4		e121.4	$$⊢\psi \to \left(\theta \to \left(\tau \to \eta \right)\right)$$
5	1	vd12	$$⊢\left(\phi ,\chi \to \psi \right)$$
6	3	vd12	$$⊢\left(\phi ,\chi \to \tau \right)$$
7	5 2 6 4	e222	$$⊢\left(\phi ,\chi \to \eta \right)$$