Theorem e1111

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 6-Mar-2012) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	e1111.1	$$⊢\left(\phi \to \psi \right)$$
	e1111.2	$$⊢\left(\phi \to \chi \right)$$
	e1111.3	$$⊢\left(\phi \to \theta \right)$$
	e1111.4	$$⊢\left(\phi \to \tau \right)$$
	e1111.5	$$⊢\psi \to \left(\chi \to \left(\theta \to \left(\tau \to \eta \right)\right)\right)$$
Assertion	e1111	$$⊢\left(\phi \to \eta \right)$$

Proof

Step	Hyp	Ref	Expression
1		e1111.1	$$⊢\left(\phi \to \psi \right)$$
2		e1111.2	$$⊢\left(\phi \to \chi \right)$$
3		e1111.3	$$⊢\left(\phi \to \theta \right)$$
4		e1111.4	$$⊢\left(\phi \to \tau \right)$$
5		e1111.5	$$⊢\psi \to \left(\chi \to \left(\theta \to \left(\tau \to \eta \right)\right)\right)$$
6	1	in1	$$⊢\phi \to \psi$$
7	2	in1	$$⊢\phi \to \chi$$
8	3	in1	$$⊢\phi \to \theta$$
9	4	in1	$$⊢\phi \to \tau$$
10	6 7 8 9 5	ee1111	$$⊢\phi \to \eta$$
11	10	dfvd1ir	$$⊢\left(\phi \to \eta \right)$$