Theorem frege18 44621

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 44603	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜑) → (𝜃 → (𝜓 → 𝜒))))
2		frege16 44619	. 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))))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-frege1 44593  ax-frege2 44594  ax-frege8 44612

This theorem is used by:  frege19  44627  frege23  44628  frege20  44631  frege51  44658  frege64a  44685  frege64b  44712  frege64c  44730  frege82  44748