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Mirrors > Home > MPE Home > Th. List > abn0OLD | Structured version Visualization version GIF version |
Description: Obsolete version of abn0 4375 as of 30-Aug-2024. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
abn0OLD | ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfab1 2899 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
2 | 1 | n0f 4337 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑}) |
3 | abid 2707 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
4 | 3 | exbii 1842 | . 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑) |
5 | 2, 4 | bitri 275 | 1 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∃wex 1773 ∈ wcel 2098 {cab 2703 ≠ wne 2934 ∅c0 4317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-dif 3946 df-nul 4318 |
This theorem is referenced by: (None) |
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