Axiom ax-frege2

Description: If a proposition ch is a necessary consequence of two propositions ps and ph and one of those, ps , is in turn a necessary consequence of the other, ph , then the proposition ch is a necessary consequence of the latter one, ph , alone. Axiom 2 of Frege1879 p. 26. Identical to ax-2 . (Contributed by RP, 24-Dec-2019) (New usage is discouraged.)

	Ref	Expression
Assertion	ax-frege2	$$⊢\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\left(\phi \to \psi \right)\to \left(\phi \to \chi \right)\right)$$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		wph	$$wff\phi$$
1		wps	$$wff\psi$$
2		wch	$$wff\chi$$
3	1 2	wi	$$wff\left(\psi \to \chi \right)$$
4	0 3	wi	$$wff\left(\phi \to \left(\psi \to \chi \right)\right)$$
5	0 1	wi	$$wff\left(\phi \to \psi \right)$$
6	0 2	wi	$$wff\left(\phi \to \chi \right)$$
7	5 6	wi	$$wff\left(\left(\phi \to \psi \right)\to \left(\phi \to \chi \right)\right)$$
8	4 7	wi	$$wff\left(\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\left(\phi \to \psi \right)\to \left(\phi \to \chi \right)\right)\right)$$