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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege59c | Structured version Visualization version GIF version |
Description: A kind of Aristotelian
inference. Proposition 59 of [Frege1879] p.
51.
Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 40157 incorrectly referenced where frege30 40176 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege59c.a | ⊢ 𝐴 ∈ 𝐵 |
Ref | Expression |
---|---|
frege59c | ⊢ ([𝐴 / 𝑥]𝜑 → (¬ [𝐴 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege59c.a | . . . 4 ⊢ 𝐴 ∈ 𝐵 | |
2 | 1 | frege58c 40265 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → [𝐴 / 𝑥](𝜑 → 𝜓)) |
3 | sbcim1 3824 | . . 3 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)) |
5 | frege30 40176 | . 2 ⊢ ((∀𝑥(𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)) → ([𝐴 / 𝑥]𝜑 → (¬ [𝐴 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑 → 𝜓)))) | |
6 | 4, 5 | ax-mp 5 | 1 ⊢ ([𝐴 / 𝑥]𝜑 → (¬ [𝐴 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1531 ∈ wcel 2110 [wsbc 3771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2173 ax-ext 2793 ax-frege1 40134 ax-frege2 40135 ax-frege8 40153 ax-frege28 40174 ax-frege58b 40245 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-v 3496 df-sbc 3772 |
This theorem is referenced by: (None) |
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