Theorem frege24 43790

Description: Closed form for a1d 25. Deduction introducing an embedded antecedent. Identical to rp-frege24 43772 which was proved without relying on ax-frege8 43784. Proposition 24 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege24	⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓)))

Proof of Theorem frege24

Step	Hyp	Ref	Expression
1		ax-frege1 43765	. 2 ⊢ ((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓)))
2		frege12 43788	. 2 ⊢ (((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓)))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 43765  ax-frege2 43766  ax-frege8 43784

This theorem is referenced by:  frege25  43792  frege63a  43856  frege63b  43883  frege63c  43901