Theorem ee13an

Description: e13an without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ee13an.1	$$⊢\phi \to \psi$$
	ee13an.2	$$⊢\phi \to \left(\chi \to \left(\theta \to \tau \right)\right)$$
	ee13an.3	$$⊢\left(\psi \wedge \tau \right)\to \eta$$
Assertion	ee13an	$$⊢\phi \to \left(\chi \to \left(\theta \to \eta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ee13an.1	$$⊢\phi \to \psi$$
2		ee13an.2	$$⊢\phi \to \left(\chi \to \left(\theta \to \tau \right)\right)$$
3		ee13an.3	$$⊢\left(\psi \wedge \tau \right)\to \eta$$
4	3	ex	$$⊢\psi \to \left(\tau \to \eta \right)$$
5	1 2 4	ee13	$$⊢\phi \to \left(\chi \to \left(\theta \to \eta \right)\right)$$