Theorem confun4

Description: An attempt at derivative. Resisted simplest path to a proof. (Contributed by Jarvin Udandy, 6-Sep-2020)

	Ref	Expression
Hypotheses	confun4.1	$$⊢\phi$$
	confun4.2	$$⊢\left(\phi \to \psi \right)\to \psi$$
	confun4.3	$$⊢\psi \to \left(\phi \to \chi \right)$$
	confun4.4	$$⊢\left(\chi \to \theta \right)\to \left(\left(\phi \to \theta \right)\leftrightarrow \psi \right)$$
	confun4.5	$$⊢\tau \leftrightarrow \left(\chi \to \theta \right)$$
	confun4.6	$$⊢\eta \leftrightarrow \neg \left(\chi \to \left(\chi \wedge \neg \chi \right)\right)$$
	confun4.7	$$⊢\psi$$
	confun4.8	$$⊢\chi \to \theta$$
Assertion	confun4	$$⊢\chi \to \left(\psi \to \tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		confun4.1	$$⊢\phi$$
2		confun4.2	$$⊢\left(\phi \to \psi \right)\to \psi$$
3		confun4.3	$$⊢\psi \to \left(\phi \to \chi \right)$$
4		confun4.4	$$⊢\left(\chi \to \theta \right)\to \left(\left(\phi \to \theta \right)\leftrightarrow \psi \right)$$
5		confun4.5	$$⊢\tau \leftrightarrow \left(\chi \to \theta \right)$$
6		confun4.6	$$⊢\eta \leftrightarrow \neg \left(\chi \to \left(\chi \wedge \neg \chi \right)\right)$$
7		confun4.7	$$⊢\psi$$
8		confun4.8	$$⊢\chi \to \theta$$
9	7 3	ax-mp	$$⊢\phi \to \chi$$
10	1 9	ax-mp	$$⊢\chi$$
11		bicom1	$$⊢\left(\tau \leftrightarrow \left(\chi \to \theta \right)\right)\to \left(\left(\chi \to \theta \right)\leftrightarrow \tau \right)$$
12	5 11	ax-mp	$$⊢\left(\chi \to \theta \right)\leftrightarrow \tau$$
13	12	biimpi	$$⊢\left(\chi \to \theta \right)\to \tau$$
14	8 13	ax-mp	$$⊢\tau$$
15	7 14	pm3.2i	$$⊢\left(\psi \wedge \tau \right)$$
16		pm3.4	$$⊢\left(\psi \wedge \tau \right)\to \left(\psi \to \tau \right)$$
17	15 16	ax-mp	$$⊢\psi \to \tau$$
18	10 17	pm3.2i	$$⊢\left(\chi \wedge \left(\psi \to \tau \right)\right)$$
19		pm3.4	$$⊢\left(\chi \wedge \left(\psi \to \tau \right)\right)\to \left(\chi \to \left(\psi \to \tau \right)\right)$$
20	18 19	ax-mp	$$⊢\chi \to \left(\psi \to \tau \right)$$