Theorem frege21 44827

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 44812	. 2 ⊢ ((𝜑 → 𝜃) → ((𝜃 → 𝜓) → (𝜑 → 𝜓)))
2		frege19 44824	. 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)))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-frege1 44790  ax-frege2 44791  ax-frege8 44809

This theorem is used by:  frege44  44848  frege47  44851