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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 40491, and ax-frege2 40492. (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 33994 | 1 wff ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) |
Colors of variables: wff setvar class |
This axiom is referenced by: frege26 40511 frege9 40513 frege12 40514 frege10 40521 frege17 40522 frege38 40542 frege53aid 40560 frege53a 40561 frege62a 40581 frege66a 40585 frege53b 40591 frege62b 40608 frege66b 40612 frege53c 40615 frege62c 40626 frege66c 40630 frege74 40638 frege84 40648 frege96 40660 |
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