Theorem frege12 40166

Description: A closed form of com23 86. Proposition 12 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege12	⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜑 → (𝜒 → (𝜓 → 𝜃))))

Proof of Theorem frege12

Step	Hyp	Ref	Expression
1		ax-frege8 40162	. 2 ⊢ ((𝜓 → (𝜒 → 𝜃)) → (𝜒 → (𝜓 → 𝜃)))
2		frege5 40153	. 2 ⊢ (((𝜓 → (𝜒 → 𝜃)) → (𝜒 → (𝜓 → 𝜃))) → ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜑 → (𝜒 → (𝜓 → 𝜃)))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜑 → (𝜒 → (𝜓 → 𝜃))))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 40143  ax-frege2 40144  ax-frege8 40162

This theorem is referenced by:  frege24  40168  frege16  40169  frege13  40175  frege15  40179  frege35  40191  frege49  40206  frege60a  40231  frege60b  40258  frege60c  40276  frege85  40301  frege127  40343