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Mirrors > Home > MPE Home > Th. List > Mathboxes > axfrege8 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
axfrege8 | ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rp-7frege 40495 | . 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))) | |
2 | rp-8frege 40498 | . 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))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-frege1 40484 ax-frege2 40485 |
This theorem is referenced by: (None) |
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