Theorem frege18 43780

Description: Closed form of a syllogism followed by a swap of antecedents. Proposition 18 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege18	⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜑) → (𝜓 → (𝜃 → 𝜒))))

Proof of Theorem frege18

Step	Hyp	Ref	Expression
1		frege5 43762	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜑) → (𝜃 → (𝜓 → 𝜒))))
2		frege16 43778	. 2 ⊢ (((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜑) → (𝜃 → (𝜓 → 𝜒)))) → ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜑) → (𝜓 → (𝜃 → 𝜒)))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜑) → (𝜓 → (𝜃 → 𝜒))))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 43752  ax-frege2 43753  ax-frege8 43771

This theorem is referenced by:  frege19  43786  frege23  43787  frege20  43790  frege51  43817  frege64a  43844  frege64b  43871  frege64c  43889  frege82  43907