Theorem ee123

Description: e123 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ee123.1	$$⊢\phi \to \psi$$
	ee123.2	$$⊢\phi \to \left(\chi \to \theta \right)$$
	ee123.3	$$⊢\phi \to \left(\chi \to \left(\tau \to \eta \right)\right)$$
	ee123.4	$$⊢\psi \to \left(\theta \to \left(\eta \to \zeta \right)\right)$$
Assertion	ee123	$$⊢\phi \to \left(\chi \to \left(\tau \to \zeta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ee123.1	$$⊢\phi \to \psi$$
2		ee123.2	$$⊢\phi \to \left(\chi \to \theta \right)$$
3		ee123.3	$$⊢\phi \to \left(\chi \to \left(\tau \to \eta \right)\right)$$
4		ee123.4	$$⊢\psi \to \left(\theta \to \left(\eta \to \zeta \right)\right)$$
5	1	a1d	$$⊢\phi \to \left(\tau \to \psi \right)$$
6	5	a1d	$$⊢\phi \to \left(\chi \to \left(\tau \to \psi \right)\right)$$
7	2	a1dd	$$⊢\phi \to \left(\chi \to \left(\tau \to \theta \right)\right)$$
8	6 7 3 4	ee333	$$⊢\phi \to \left(\chi \to \left(\tau \to \zeta \right)\right)$$