confun4

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

	Ref	Expression
Hypotheses	confun4.1	$⊢ φ$
	confun4.2	$⊢ (φ \to ψ) \to ψ$
	confun4.3	$⊢ ψ \to (φ \to χ)$
	confun4.4	$⊢ (χ \to θ) \to ((φ \to θ) \leftrightarrow ψ)$
	confun4.5	$⊢ τ \leftrightarrow (χ \to θ)$
	confun4.6	$⊢ η \leftrightarrow \neg (χ \to (χ \land \neg χ))$
	confun4.7	$⊢ ψ$
	confun4.8	$⊢ χ \to θ$
Assertion	confun4	$⊢ χ \to (ψ \to τ)$

Step	Hyp	Ref	Expression
1		confun4.1	$⊢ φ$
2		confun4.2	$⊢ (φ \to ψ) \to ψ$
3		confun4.3	$⊢ ψ \to (φ \to χ)$
4		confun4.4	$⊢ (χ \to θ) \to ((φ \to θ) \leftrightarrow ψ)$
5		confun4.5	$⊢ τ \leftrightarrow (χ \to θ)$
6		confun4.6	$⊢ η \leftrightarrow \neg (χ \to (χ \land \neg χ))$
7		confun4.7	$⊢ ψ$
8		confun4.8	$⊢ χ \to θ$
9	7 3	ax-mp	$⊢ φ \to χ$
10	1 9	ax-mp	$⊢ χ$
11		bicom1	$⊢ (τ \leftrightarrow (χ \to θ)) \to ((χ \to θ) \leftrightarrow τ)$
12	5 11	ax-mp	$⊢ (χ \to θ) \leftrightarrow τ$
13	12	biimpi	$⊢ (χ \to θ) \to τ$
14	8 13	ax-mp	$⊢ τ$
15	7 14	pm3.2i	$⊢ (ψ \land τ)$
16		pm3.4	$⊢ (ψ \land τ) \to (ψ \to τ)$
17	15 16	ax-mp	$⊢ ψ \to τ$
18	10 17	pm3.2i	$⊢ (χ \land (ψ \to τ))$
19		pm3.4	$⊢ (χ \land (ψ \to τ)) \to (χ \to (ψ \to τ))$
20	18 19	ax-mp	$⊢ χ \to (ψ \to τ)$

Metamath Proof Explorer