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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege62c | Structured version Visualization version GIF version |
Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2743 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege59c.a | ⊢ 𝐴 ∈ 𝐵 |
Ref | Expression |
---|---|
frege62c | ⊢ ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [𝐴 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege59c.a | . . . 4 ⊢ 𝐴 ∈ 𝐵 | |
2 | 1 | frege58c 40145 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → [𝐴 / 𝑥](𝜑 → 𝜓)) |
3 | sbcim1 3822 | . . 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 | ax-frege8 40033 | . 2 ⊢ ((∀𝑥(𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)) → ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [𝐴 / 𝑥]𝜓))) | |
6 | 4, 5 | ax-mp 5 | 1 ⊢ ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [𝐴 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 ∈ wcel 2105 [wsbc 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2167 ax-ext 2790 ax-frege8 40033 ax-frege58b 40125 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-v 3494 df-sbc 3770 |
This theorem is referenced by: frege63c 40150 frege64c 40151 |
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