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| 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 3002 | . 2 ⊢ (𝐴 = if(𝐴 ≠ 0, 𝐴, 1) → (𝐴 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0)) | |
| 2 | neeq1 3002 | . 2 ⊢ (1 = if(𝐴 ≠ 0, 𝐴, 1) → (1 ≠ 0 ↔ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0)) | |
| 3 | ax-1ne0 11225 | . 2 ⊢ 1 ≠ 0 | |
| 4 | 1, 2, 3 | elimhyp 4590 | 1 ⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ≠ wne 2939 ifcif 4524 0cc0 11156 1c1 11157 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-1ne0 11225 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-if 4525 | 
| This theorem is referenced by: sqdivzi 35729 | 
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