Theorem frege19 44272

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 44260	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜒 → 𝜃) → (𝜓 → 𝜃)))
2		frege18 44266	. 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 44238  ax-frege2 44239  ax-frege8 44257

This theorem is referenced by:  frege21  44275  frege20  44276  frege71  44382  frege86  44397  frege103  44414  frege119  44430  frege123  44434