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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 43826 incorrectly referenced where frege30 43845 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 43934 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → [𝐴 / 𝑥](𝜑 → 𝜓)) | 
| 3 | sbcim1 3842 | . . 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 43845 | . 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 1538 ∈ wcel 2108 [wsbc 3788 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-frege1 43803 ax-frege2 43804 ax-frege8 43822 ax-frege28 43843 ax-frege58b 43914 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sbc 3789 | 
| This theorem is referenced by: (None) | 
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