Theorem ee233

Description: Non-virtual deduction form of e233 . (Contributed by Alan Sare, 18-Mar-2012) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.

	Ref	Expression
Hypotheses	ee233.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	ee233.2	$$⊢\phi \to \left(\psi \to \left(\theta \to \tau \right)\right)$$
	ee233.3	$$⊢\phi \to \left(\psi \to \left(\theta \to \eta \right)\right)$$
	ee233.4	$$⊢\chi \to \left(\tau \to \left(\eta \to \zeta \right)\right)$$
Assertion	ee233	$$⊢\phi \to \left(\psi \to \left(\theta \to \zeta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ee233.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		ee233.2	$$⊢\phi \to \left(\psi \to \left(\theta \to \tau \right)\right)$$
3		ee233.3	$$⊢\phi \to \left(\psi \to \left(\theta \to \eta \right)\right)$$
4		ee233.4	$$⊢\chi \to \left(\tau \to \left(\eta \to \zeta \right)\right)$$
5	1 4	syl6	$$⊢\phi \to \left(\psi \to \left(\tau \to \left(\eta \to \zeta \right)\right)\right)$$
6	5	com3r	$$⊢\tau \to \left(\phi \to \left(\psi \to \left(\eta \to \zeta \right)\right)\right)$$
7	2 6	syl8	$$⊢\phi \to \left(\psi \to \left(\theta \to \left(\phi \to \left(\psi \to \left(\eta \to \zeta \right)\right)\right)\right)\right)$$
8		pm2.43cbi	$$⊢\left(\phi \to \left(\psi \to \left(\theta \to \left(\phi \to \left(\psi \to \left(\eta \to \zeta \right)\right)\right)\right)\right)\right)\leftrightarrow \left(\psi \to \left(\theta \to \left(\phi \to \left(\psi \to \left(\eta \to \zeta \right)\right)\right)\right)\right)$$
9	7 8	mpbi	$$⊢\psi \to \left(\theta \to \left(\phi \to \left(\psi \to \left(\eta \to \zeta \right)\right)\right)\right)$$
10		pm2.43cbi	$$⊢\left(\psi \to \left(\theta \to \left(\phi \to \left(\psi \to \left(\eta \to \zeta \right)\right)\right)\right)\right)\leftrightarrow \left(\theta \to \left(\phi \to \left(\psi \to \left(\eta \to \zeta \right)\right)\right)\right)$$
11	9 10	mpbi	$$⊢\theta \to \left(\phi \to \left(\psi \to \left(\eta \to \zeta \right)\right)\right)$$
12	11	com14	$$⊢\eta \to \left(\phi \to \left(\psi \to \left(\theta \to \zeta \right)\right)\right)$$
13	3 12	syl8	$$⊢\phi \to \left(\psi \to \left(\theta \to \left(\phi \to \left(\psi \to \left(\theta \to \zeta \right)\right)\right)\right)\right)$$
14		pm2.43cbi	$$⊢\left(\phi \to \left(\psi \to \left(\theta \to \left(\phi \to \left(\psi \to \left(\theta \to \zeta \right)\right)\right)\right)\right)\right)\leftrightarrow \left(\psi \to \left(\theta \to \left(\phi \to \left(\psi \to \left(\theta \to \zeta \right)\right)\right)\right)\right)$$
15	13 14	mpbi	$$⊢\psi \to \left(\theta \to \left(\phi \to \left(\psi \to \left(\theta \to \zeta \right)\right)\right)\right)$$
16		pm2.43cbi	$$⊢\left(\psi \to \left(\theta \to \left(\phi \to \left(\psi \to \left(\theta \to \zeta \right)\right)\right)\right)\right)\leftrightarrow \left(\theta \to \left(\phi \to \left(\psi \to \left(\theta \to \zeta \right)\right)\right)\right)$$
17	15 16	mpbi	$$⊢\theta \to \left(\phi \to \left(\psi \to \left(\theta \to \zeta \right)\right)\right)$$
18		pm2.43cbi	$$⊢\left(\theta \to \left(\phi \to \left(\psi \to \left(\theta \to \zeta \right)\right)\right)\right)\leftrightarrow \left(\phi \to \left(\psi \to \left(\theta \to \zeta \right)\right)\right)$$
19	17 18	mpbi	$$⊢\phi \to \left(\psi \to \left(\theta \to \zeta \right)\right)$$