Theorem confun3

Description: Confun's more complex form where both a,d have been "defined". (Contributed by Jarvin Udandy, 6-Sep-2020)

	Ref	Expression
Hypotheses	confun3.1	$$⊢\phi \leftrightarrow \left(\chi \to \psi \right)$$
	confun3.2	$$⊢\theta \leftrightarrow \neg \left(\chi \to \left(\chi \wedge \neg \chi \right)\right)$$
	confun3.3	$$⊢\chi \to \psi$$
	confun3.4	$$⊢\chi \to \neg \left(\chi \to \left(\chi \wedge \neg \chi \right)\right)$$
	confun3.5	$$⊢\left(\chi \to \psi \right)\to \left(\left(\chi \to \psi \right)\to \psi \right)$$
Assertion	confun3	$$⊢\chi \to \left(\neg \left(\chi \to \left(\chi \wedge \neg \chi \right)\right)\leftrightarrow \left(\chi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		confun3.1	$$⊢\phi \leftrightarrow \left(\chi \to \psi \right)$$
2		confun3.2	$$⊢\theta \leftrightarrow \neg \left(\chi \to \left(\chi \wedge \neg \chi \right)\right)$$
3		confun3.3	$$⊢\chi \to \psi$$
4		confun3.4	$$⊢\chi \to \neg \left(\chi \to \left(\chi \wedge \neg \chi \right)\right)$$
5		confun3.5	$$⊢\left(\chi \to \psi \right)\to \left(\left(\chi \to \psi \right)\to \psi \right)$$
6	3 3 4 5	confun	$$⊢\chi \to \left(\neg \left(\chi \to \left(\chi \wedge \neg \chi \right)\right)\leftrightarrow \left(\chi \to \psi \right)\right)$$