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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.) Avoid ax-10 2141, ax-12 2177. (Revised by GG, 28-Jun-2024.) | 
| Ref | Expression | 
|---|---|
| rspn0 | ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | n0 4353 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
| 2 | df-ral 3062 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
| 3 | exim 1834 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥𝜑)) | |
| 4 | ax5e 1912 | . . . 4 ⊢ (∃𝑥𝜑 → 𝜑) | |
| 5 | 3, 4 | syl6com 37 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → 𝜑)) | 
| 6 | 2, 5 | biimtrid 242 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) | 
| 7 | 1, 6 | sylbi 217 | 1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∅c0 4333 | 
| 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-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-ne 2941 df-ral 3062 df-dif 3954 df-nul 4334 | 
| This theorem is referenced by: hashge2el2dif 14519 rmodislmodlem 20927 rmodislmod 20928 scmatf1 22537 fusgrregdegfi 29587 rusgr1vtxlem 29605 upgrewlkle2 29624 zarclsiin 33870 ralralimp 47290 | 
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