Theorem frege5 43790

Description: A closed form of syl 17. Identical to imim2 58. Theorem *2.05 of [WhiteheadRussell] p. 100. Proposition 5 of [Frege1879] p. 32. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege5	⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))

Proof of Theorem frege5

Step	Hyp	Ref	Expression
1		ax-frege1 43780	. 2 ⊢ ((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓)))
2		frege4 43789	. 2 ⊢ (((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 43780  ax-frege2 43781

This theorem is referenced by:  rp-frege25  43795  frege6  43796  frege7  43798  frege9  43802  frege12  43803  frege16  43806  frege25  43807  frege18  43808  frege22  43809  frege14  43813  frege29  43821  frege34  43827  frege45  43839  frege80  43933  frege90  43943