Theorem frege21 40193

Description: Replace antecedent in antecedent. Proposition 21 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege21	⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜑 → 𝜃) → ((𝜃 → 𝜓) → 𝜒)))

Proof of Theorem frege21

Step	Hyp	Ref	Expression
1		frege9 40178	. 2 ⊢ ((𝜑 → 𝜃) → ((𝜃 → 𝜓) → (𝜑 → 𝜓)))
2		frege19 40190	. 2 ⊢ (((𝜑 → 𝜃) → ((𝜃 → 𝜓) → (𝜑 → 𝜓))) → (((𝜑 → 𝜓) → 𝜒) → ((𝜑 → 𝜃) → ((𝜃 → 𝜓) → 𝜒))))
3	1, 2	ax-mp 5	1 ⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜑 → 𝜃) → ((𝜃 → 𝜓) → 𝜒)))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 40156  ax-frege2 40157  ax-frege8 40175

This theorem is referenced by:  frege44  40214  frege47  40217