Theorem frege24 44642

Description: Closed form for a1d 26. Deduction introducing an embedded antecedent. Identical to rp-frege24 44624 which was proved without relying on ax-frege8 44636. 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 44617	. 2 ⊢ ((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓)))
2		frege12 44640	. 2 ⊢ (((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓)))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-frege1 44617  ax-frege2 44618  ax-frege8 44636

This theorem is used by:  frege25  44644  frege63a  44708  frege63b  44735  frege63c  44753