Theorem frege5 44799

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 44789	. 2 ⊢ ((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓)))
2		frege4 44798	. 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 44789  ax-frege2 44790

This theorem is used by:  rp-frege25  44804  frege6  44805  frege7  44807  frege9  44811  frege12  44812  frege16  44815  frege25  44816  frege18  44817  frege22  44818  frege14  44822  frege29  44830  frege34  44836  frege45  44848  frege80  44942  frege90  44952