Axiom ax-frege2 40143

Description: If a proposition 𝜒 is a necessary consequence of two propositions 𝜓 and 𝜑 and one of those, 𝜓, is in turn a necessary consequence of the other, 𝜑, then the proposition 𝜒 is a necessary consequence of the latter one, 𝜑, alone. Axiom 2 of [Frege1879] p. 26. Identical to ax-2 7. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-frege2	⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))

Detailed syntax breakdown of Axiom ax-frege2

Step	Hyp	Ref	Expression
1		wph	. 2 wff 𝜑
2		wps	. . 3 wff 𝜓
3		wch	. . 3 wff 𝜒
4	2, 3	wi 4	. 2 wff (𝜓 → 𝜒)
5	1, 3	wi 4	. . 3 wff (𝜑 → 𝜒)
6	1, 2, 5	bj-0 33885	. 2 wff ((𝜑 → 𝜓) → (𝜑 → 𝜒))
7	1, 4, 6	bj-0 33885	1 wff ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))

This axiom is referenced by:  rp-frege3g  40146  frege3  40147  rp-misc1-frege  40148  rp-frege24  40149  rp-frege4g  40150  frege4  40151  rp-7frege  40153  rp-8frege  40156  frege39  40194  frege73  40288  frege79  40294