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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 2139, ax-12 2175. (Revised by GG, 28-Jun-2024.) |
Ref | Expression |
---|---|
rspn0 | ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4359 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | df-ral 3060 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
3 | exim 1831 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥𝜑)) | |
4 | ax5e 1910 | . . . 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 1535 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-ne 2939 df-ral 3060 df-dif 3966 df-nul 4340 |
This theorem is referenced by: hashge2el2dif 14516 rmodislmodlem 20944 rmodislmod 20945 rmodislmodOLD 20946 scmatf1 22553 fusgrregdegfi 29602 rusgr1vtxlem 29620 upgrewlkle2 29639 zarclsiin 33832 ralralimp 47228 |
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