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Mirrors > Home > MPE Home > Th. List > elimne0 | Structured version Visualization version GIF version |
Description: Hypothesis for weak deduction theorem to eliminate 𝐴 ≠ 0. (Contributed by NM, 15-May-1999.) |
Ref | Expression |
---|---|
elimne0 | ⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq1 2844 | . 2 ⊢ (𝐴 = if(𝐴 ≠ 0, 𝐴, 1) → (𝐴 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0)) | |
2 | neeq1 2844 | . 2 ⊢ (1 = if(𝐴 ≠ 0, 𝐴, 1) → (1 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0)) | |
3 | ax-1ne0 9884 | . 2 ⊢ 1 ≠ 0 | |
4 | 1, 2, 3 | elimhyp 4096 | 1 ⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2780 ifcif 4036 0cc0 9815 1c1 9816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-1ne0 9884 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-ne 2782 df-if 4037 |
This theorem is referenced by: sqdivzi 30863 |
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