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| Mirrors > Home > MPE Home > Th. List > Mathboxes > frege59b | Structured version Visualization version GIF version | ||
| Description: A kind of Aristotelian
inference.  Namely Felapton or Fesapo.  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 43831 incorrectly referenced where frege30 43850 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| frege59b | ⊢ ([𝑦 / 𝑥]𝜑 → (¬ [𝑦 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑 → 𝜓))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | frege58bcor 43921 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
| 2 | frege30 43850 | . 2 ⊢ ((∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) → ([𝑦 / 𝑥]𝜑 → (¬ [𝑦 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑 → 𝜓)))) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → (¬ [𝑦 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑 → 𝜓))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∀wal 1537 [wsb 2063 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-10 2140 ax-12 2176 ax-frege1 43808 ax-frege2 43809 ax-frege8 43827 ax-frege28 43848 ax-frege58b 43919 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-nf 1783 df-sb 2064 | 
| This theorem is referenced by: (None) | 
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