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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 4224 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | nfra1 3184 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑 | |
3 | nfv 1890 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | 2, 3 | nfim 1876 | . . 3 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐴 𝜑 → 𝜑) |
5 | rsp 3170 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑)) | |
6 | 5 | com12 32 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
7 | 4, 6 | exlimi 2180 | . 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 1759 ∈ wcel 2079 ≠ wne 2982 ∀wral 3103 ∅c0 4206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-ext 2767 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-dif 3857 df-nul 4207 |
This theorem is referenced by: hashge2el2dif 13672 rmodislmodlem 19379 rmodislmod 19380 scmatf1 20812 fusgrregdegfi 27022 rusgr1vtxlem 27040 upgrewlkle2 27059 ralralimp 42947 |
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