Theorem frege19 43815

Description: A closed form of syl6 35. Proposition 19 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege19	⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜒 → 𝜃) → (𝜑 → (𝜓 → 𝜃))))

Proof of Theorem frege19

Step	Hyp	Ref	Expression
1		frege9 43803	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜒 → 𝜃) → (𝜓 → 𝜃)))
2		frege18 43809	. 2 ⊢ (((𝜓 → 𝜒) → ((𝜒 → 𝜃) → (𝜓 → 𝜃))) → ((𝜑 → (𝜓 → 𝜒)) → ((𝜒 → 𝜃) → (𝜑 → (𝜓 → 𝜃)))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜒 → 𝜃) → (𝜑 → (𝜓 → 𝜃))))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 43781  ax-frege2 43782  ax-frege8 43800

This theorem is referenced by:  frege21  43818  frege20  43819  frege71  43925  frege86  43940  frege103  43957  frege119  43973  frege123  43977