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Mirrors > Home > MPE Home > Th. List > rspn0OLD | Structured version Visualization version GIF version |
Description: Obsolete version of rspn0 4245 as of 28-Jun-2024. (Contributed by Alexander van der Vekens, 6-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rspn0OLD | ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4239 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | nfra1 3145 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑 | |
3 | nfv 1916 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | 2, 3 | nfim 1898 | . . 3 ⊢ Ⅎ𝑥(∀𝑥 ∈ 𝐴 𝜑 → 𝜑) |
5 | rsp 3132 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑)) | |
6 | 5 | com12 32 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
7 | 4, 6 | exlimi 2216 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
8 | 1, 7 | sylbi 220 | 1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1782 ∈ wcel 2112 ≠ wne 2949 ∀wral 3068 ∅c0 4221 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-12 2176 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-ne 2950 df-ral 3073 df-dif 3857 df-nul 4222 |
This theorem is referenced by: (None) |
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