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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 3080 | . 2 ⊢ (𝐴 = if(𝐴 ≠ 0, 𝐴, 1) → (𝐴 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0)) | |
2 | neeq1 3080 | . 2 ⊢ (1 = if(𝐴 ≠ 0, 𝐴, 1) → (1 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0)) | |
3 | ax-1ne0 10608 | . 2 ⊢ 1 ≠ 0 | |
4 | 1, 2, 3 | elimhyp 4532 | 1 ⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 3018 ifcif 4469 0cc0 10539 1c1 10540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2795 ax-1ne0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-ne 3019 df-if 4470 |
This theorem is referenced by: sqdivzi 32961 |
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