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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 3004 | . 2 ⊢ (𝐴 = if(𝐴 ≠ 0, 𝐴, 1) → (𝐴 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0)) | |
2 | neeq1 3004 | . 2 ⊢ (1 = if(𝐴 ≠ 0, 𝐴, 1) → (1 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0)) | |
3 | ax-1ne0 11179 | . 2 ⊢ 1 ≠ 0 | |
4 | 1, 2, 3 | elimhyp 4594 | 1 ⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2941 ifcif 4529 0cc0 11110 1c1 11111 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-1ne0 11179 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-if 4530 |
This theorem is referenced by: sqdivzi 34697 |
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