Theorem frege10 40172

Description: Result commuting antecedents within an antecedent. Proposition 10 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege10	⊢ (((𝜑 → (𝜓 → 𝜒)) → 𝜃) → ((𝜓 → (𝜑 → 𝜒)) → 𝜃))

Proof of Theorem frege10

Step	Hyp	Ref	Expression
1		ax-frege8 40161	. 2 ⊢ ((𝜓 → (𝜑 → 𝜒)) → (𝜑 → (𝜓 → 𝜒)))
2		frege9 40164	. 2 ⊢ (((𝜓 → (𝜑 → 𝜒)) → (𝜑 → (𝜓 → 𝜒))) → (((𝜑 → (𝜓 → 𝜒)) → 𝜃) → ((𝜓 → (𝜑 → 𝜒)) → 𝜃)))
3	1, 2	ax-mp 5	1 ⊢ (((𝜑 → (𝜓 → 𝜒)) → 𝜃) → ((𝜓 → (𝜑 → 𝜒)) → 𝜃))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 40142  ax-frege2 40143  ax-frege8 40161

This theorem is referenced by:  frege30  40184