Theorem frege20 44904

Description: A closed form of syl8 77. Proposition 20 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege20	⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → ((𝜃 → 𝜏) → (𝜑 → (𝜓 → (𝜒 → 𝜏)))))

Proof of Theorem frege20

Step	Hyp	Ref	Expression
1		frege19 44900	. 2 ⊢ ((𝜓 → (𝜒 → 𝜃)) → ((𝜃 → 𝜏) → (𝜓 → (𝜒 → 𝜏))))
2		frege18 44894	. 2 ⊢ (((𝜓 → (𝜒 → 𝜃)) → ((𝜃 → 𝜏) → (𝜓 → (𝜒 → 𝜏)))) → ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → ((𝜃 → 𝜏) → (𝜑 → (𝜓 → (𝜒 → 𝜏))))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → ((𝜃 → 𝜏) → (𝜑 → (𝜓 → (𝜒 → 𝜏)))))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-frege1 44866  ax-frege2 44867  ax-frege8 44885

This theorem is used by:  frege121  45060  frege125  45064