Theorem ee31an

Description: e31an without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ee31an.1	$$⊢\phi \to \left(\psi \to \left(\chi \to \theta \right)\right)$$
	ee31an.2	$$⊢\phi \to \tau$$
	ee31an.3	$$⊢\left(\theta \wedge \tau \right)\to \eta$$
Assertion	ee31an	$$⊢\phi \to \left(\psi \to \left(\chi \to \eta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ee31an.1	$$⊢\phi \to \left(\psi \to \left(\chi \to \theta \right)\right)$$
2		ee31an.2	$$⊢\phi \to \tau$$
3		ee31an.3	$$⊢\left(\theta \wedge \tau \right)\to \eta$$
4	2	a1d	$$⊢\phi \to \left(\chi \to \tau \right)$$
5	4	a1d	$$⊢\phi \to \left(\psi \to \left(\chi \to \tau \right)\right)$$
6	1 5 3	ee33an	$$⊢\phi \to \left(\psi \to \left(\chi \to \eta \right)\right)$$