Theorem frege19 37138

Description: A closed form of syl6 34. 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 37126	. 2 ⊢ ((𝜓 → 𝜒) → ((𝜒 → 𝜃) → (𝜓 → 𝜃)))
2		frege18 37132	. 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 37104  ax-frege2 37105  ax-frege8 37123

This theorem is referenced by:  frege21  37141  frege20  37142  frege71  37248  frege86  37263  frege103  37280  frege119  37296  frege123  37300