Theorem frege19 44664

Description: A closed form of syl6 36. 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 44652	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜒 → 𝜃) → (𝜓 → 𝜃)))
2		frege18 44658	. 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))))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-frege1 44630  ax-frege2 44631  ax-frege8 44649

This theorem is used by:  frege21  44667  frege20  44668  frege71  44774  frege86  44789  frege103  44806  frege119  44822  frege123  44826