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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 42159 incorrectly referenced where frege30 42178 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 42267 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → [𝐴 / 𝑥](𝜑 → 𝜓)) |
3 | sbcim1 3800 | . . 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 42178 | . 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 1540 ∈ wcel 2107 [wsbc 3744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-frege1 42136 ax-frege2 42137 ax-frege8 42155 ax-frege28 42176 ax-frege58b 42247 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-sbc 3745 |
This theorem is referenced by: (None) |
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