![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4131 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | nfra1 3122 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑 | |
3 | nfv 2010 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | 2, 3 | nfim 1996 | . . 3 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐴 𝜑 → 𝜑) |
5 | rsp 3110 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑)) | |
6 | 5 | com12 32 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
7 | 4, 6 | exlimi 2252 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
8 | 1, 7 | sylbi 209 | 1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1875 ∈ wcel 2157 ≠ wne 2971 ∀wral 3089 ∅c0 4115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-v 3387 df-dif 3772 df-nul 4116 |
This theorem is referenced by: hashge2el2dif 13511 rmodislmodlem 19248 rmodislmod 19249 scmatf1 20663 fusgrregdegfi 26819 rusgr1vtxlem 26837 upgrewlkle2 26856 ralralimp 42133 |
Copyright terms: Public domain | W3C validator |