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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax-frege8 | Structured version Visualization version GIF version |
Description: Swap antecedents. If two conditions have a proposition as a consequence, their order is immaterial. Third axiom of Frege's 1879 work but identical to pm2.04 90 which can be proved from only ax-mp 5, ax-frege1 42312, and ax-frege2 42313. (Redundant) Axiom 8 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax-frege8 | ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . 2 wff 𝜑 | |
2 | wps | . . 3 wff 𝜓 | |
3 | wch | . . 3 wff 𝜒 | |
4 | 2, 3 | wi 4 | . 2 wff (𝜓 → 𝜒) |
5 | 1, 3 | wi 4 | . . 3 wff (𝜑 → 𝜒) |
6 | 2, 5 | wi 4 | . 2 wff (𝜓 → (𝜑 → 𝜒)) |
7 | 1, 4, 6 | bj-0 35222 | 1 wff ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) |
Colors of variables: wff setvar class |
This axiom is referenced by: frege26 42332 frege9 42334 frege12 42335 frege10 42342 frege17 42343 frege38 42363 frege53aid 42381 frege53a 42382 frege62a 42402 frege66a 42406 frege53b 42412 frege62b 42429 frege66b 42433 frege53c 42436 frege62c 42447 frege66c 42451 frege74 42459 frege84 42469 frege96 42481 |
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