Theorem ee102

Description: e102 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ee102.1	$$⊢\phi \to \psi$$
	ee102.2	$$⊢\chi$$
	ee102.3	$$⊢\phi \to \left(\theta \to \tau \right)$$
	ee102.4	$$⊢\psi \to \left(\chi \to \left(\tau \to \eta \right)\right)$$
Assertion	ee102	$$⊢\phi \to \left(\theta \to \eta \right)$$

Proof

Step	Hyp	Ref	Expression
1		ee102.1	$$⊢\phi \to \psi$$
2		ee102.2	$$⊢\chi$$
3		ee102.3	$$⊢\phi \to \left(\theta \to \tau \right)$$
4		ee102.4	$$⊢\psi \to \left(\chi \to \left(\tau \to \eta \right)\right)$$
5	1	a1d	$$⊢\phi \to \left(\theta \to \psi \right)$$
6	2	a1i	$$⊢\theta \to \chi$$
7	6	a1i	$$⊢\phi \to \left(\theta \to \chi \right)$$
8	5 7 3 4	ee222	$$⊢\phi \to \left(\theta \to \eta \right)$$