confun5

Description: An attempt at derivative. Resisted simplest path to a proof. Interesting that ch, th, ta, et were all provable. (Contributed by Jarvin Udandy, 7-Sep-2020)

	Ref	Expression
Hypotheses	confun5.1	$⊢ φ$
	confun5.2	$⊢ (φ \to ψ) \to ψ$
	confun5.3	$⊢ ψ \to (φ \to χ)$
	confun5.4	$⊢ (χ \to θ) \to ((φ \to θ) \leftrightarrow ψ)$
	confun5.5	$⊢ τ \leftrightarrow (χ \to θ)$
	confun5.6	$⊢ η \leftrightarrow \neg (χ \to (χ \land \neg χ))$
	confun5.7	$⊢ ψ$
	confun5.8	$⊢ χ \to θ$
Assertion	confun5	$⊢ χ \to (η \leftrightarrow τ)$

Proof

Step	Hyp	Ref	Expression
1		confun5.1	$⊢ φ$
2		confun5.2	$⊢ (φ \to ψ) \to ψ$
3		confun5.3	$⊢ ψ \to (φ \to χ)$
4		confun5.4	$⊢ (χ \to θ) \to ((φ \to θ) \leftrightarrow ψ)$
5		confun5.5	$⊢ τ \leftrightarrow (χ \to θ)$
6		confun5.6	$⊢ η \leftrightarrow \neg (χ \to (χ \land \neg χ))$
7		confun5.7	$⊢ ψ$
8		confun5.8	$⊢ χ \to θ$
9	7 3	ax-mp	$⊢ φ \to χ$
10	1 9	ax-mp	$⊢ χ$
11	10	atnaiana	$⊢ \neg (χ \to (χ \land \neg χ))$
12		bicom1	$⊢ (η \leftrightarrow \neg (χ \to (χ \land \neg χ))) \to (\neg (χ \to (χ \land \neg χ)) \leftrightarrow η)$
13	6 12	ax-mp	$⊢ \neg (χ \to (χ \land \neg χ)) \leftrightarrow η$
14	13	biimpi	$⊢ \neg (χ \to (χ \land \neg χ)) \to η$
15	11 14	ax-mp	$⊢ η$
16		bicom1	$⊢ (τ \leftrightarrow (χ \to θ)) \to ((χ \to θ) \leftrightarrow τ)$
17	5 16	ax-mp	$⊢ (χ \to θ) \leftrightarrow τ$
18	17	biimpi	$⊢ (χ \to θ) \to τ$
19	8 18	ax-mp	$⊢ τ$
20	15 19	2th	$⊢ η \leftrightarrow τ$
21		ax-1	$⊢ (η \leftrightarrow τ) \to (χ \to (η \leftrightarrow τ))$
22	20 21	ax-mp	$⊢ χ \to (η \leftrightarrow τ)$

Metamath Proof Explorer

Theorem confun5

Proof