Theorem frege5 44943

Description: A closed form of syl 18. Identical to imim2 59. 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 44933	. 2 ⊢ ((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓)))
2		frege4 44942	. 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)))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-frege1 44933  ax-frege2 44934

This theorem is used by:  rp-frege25  44948  frege6  44949  frege7  44951  frege9  44955  frege12  44956  frege16  44959  frege25  44960  frege18  44961  frege22  44962  frege14  44966  frege29  44974  frege34  44980  frege45  44992  frege80  45086  frege90  45096