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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 40129, and ax-frege2 40130. (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 33876 | 1 wff ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) |
Colors of variables: wff setvar class |
This axiom is referenced by: frege26 40149 frege9 40151 frege12 40152 frege10 40159 frege17 40160 frege38 40180 frege53aid 40198 frege53a 40199 frege62a 40219 frege66a 40223 frege53b 40229 frege62b 40246 frege66b 40250 frege53c 40253 frege62c 40264 frege66c 40268 frege74 40276 frege84 40286 frege96 40298 |
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