Theorem frege24 44388

Description: Closed form for a1d 25. Deduction introducing an embedded antecedent. Identical to rp-frege24 44370 which was proved without relying on ax-frege8 44382. 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 44363	. 2 ⊢ ((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓)))
2		frege12 44386	. 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 44363  ax-frege2 44364  ax-frege8 44382

This theorem is referenced by:  frege25  44390  frege63a  44454  frege63b  44481  frege63c  44499