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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 2138, ax-12 2172. (Revised by Gino Giotto, 28-Jun-2024.) |
Ref | Expression |
---|---|
rspn0 | ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4307 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | df-ral 3062 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
3 | exim 1837 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥𝜑)) | |
4 | ax5e 1916 | . . . 4 ⊢ (∃𝑥𝜑 → 𝜑) | |
5 | 3, 4 | syl6com 37 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → 𝜑)) |
6 | 2, 5 | biimtrid 241 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
7 | 1, 6 | sylbi 216 | 1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∃wex 1782 ∈ wcel 2107 ≠ wne 2940 ∀wral 3061 ∅c0 4283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-ne 2941 df-ral 3062 df-dif 3914 df-nul 4284 |
This theorem is referenced by: hashge2el2dif 14385 rmodislmodlem 20404 rmodislmod 20405 rmodislmodOLD 20406 scmatf1 21896 fusgrregdegfi 28559 rusgr1vtxlem 28577 upgrewlkle2 28596 zarclsiin 32509 ralralimp 45596 |
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