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Mirrors > Home > MPE Home > Th. List > rspn0 | Structured version Visualization version GIF version |
Description: Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.) |
Ref | Expression |
---|---|
rspn0 | ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4307 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | nfra1 3216 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑 | |
3 | nfv 1906 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | 2, 3 | nfim 1888 | . . 3 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐴 𝜑 → 𝜑) |
5 | rsp 3202 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑)) | |
6 | 5 | com12 32 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
7 | 4, 6 | exlimi 2207 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
8 | 1, 7 | sylbi 218 | 1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1771 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∅c0 4288 |
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-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-dif 3936 df-nul 4289 |
This theorem is referenced by: hashge2el2dif 13826 rmodislmodlem 19630 rmodislmod 19631 scmatf1 21068 fusgrregdegfi 27278 rusgr1vtxlem 27296 upgrewlkle2 27315 ralralimp 43354 |
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