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| Description: Consequence of "restricted at most one". (Contributed by Thierry Arnoux, 9-Dec-2019.) | 
| Ref | Expression | 
|---|---|
| rmoi2.1 | ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒)) | 
| rmoi2.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐴) | 
| rmoi2.3 | ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) | 
| rmoi2.4 | ⊢ (𝜑 → 𝑥 ∈ 𝐴) | 
| rmoi2.5 | ⊢ (𝜑 → 𝜓) | 
| rmoi2.6 | ⊢ (𝜑 → 𝜒) | 
| Ref | Expression | 
|---|---|
| rmoi2 | ⊢ (𝜑 → 𝑥 = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rmoi2.6 | . 2 ⊢ (𝜑 → 𝜒) | |
| 2 | rmoi2.1 | . . 3 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒)) | |
| 3 | rmoi2.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
| 4 | rmoi2.3 | . . 3 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) | |
| 5 | rmoi2.4 | . . 3 ⊢ (𝜑 → 𝑥 ∈ 𝐴) | |
| 6 | rmoi2.5 | . . 3 ⊢ (𝜑 → 𝜓) | |
| 7 | 2, 3, 4, 5, 6 | rmob2 3891 | . 2 ⊢ (𝜑 → (𝑥 = 𝐵 ↔ 𝜒)) | 
| 8 | 1, 7 | mpbird 257 | 1 ⊢ (𝜑 → 𝑥 = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 ∃*wrmo 3378 | 
| 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-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-rmo 3379 df-v 3481 | 
| This theorem is referenced by: lmieu 28793 | 
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