Theorem frege5 44377

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 44367	. 2 ⊢ ((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓)))
2		frege4 44376	. 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 44367  ax-frege2 44368

This theorem is referenced by:  rp-frege25  44382  frege6  44383  frege7  44385  frege9  44389  frege12  44390  frege16  44393  frege25  44394  frege18  44395  frege22  44396  frege14  44400  frege29  44408  frege34  44414  frege45  44426  frege80  44520  frege90  44530