Theorem axfrege8 41415

Description: Swap antecedents. Identical to pm2.04 90. This demonstrates that Axiom 8 of [Frege1879] p. 35 is redundant.

Proof follows closely proof of pm2.04 90 in https://us.metamath.org/mmsolitaire/pmproofs.txt 90, but in the style of Frege's 1879 work. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
axfrege8	⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))

Proof of Theorem axfrege8

Step	Hyp	Ref	Expression
1		rp-7frege 41409	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → ((𝜑 → 𝜓) → (𝜑 → 𝜒))))
2		rp-8frege 41412	. 2 ⊢ (((𝜑 → (𝜓 → 𝜒)) → (𝜓 → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))) → ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 41398  ax-frege2 41399

This theorem is referenced by: (None)