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| Description: Condition implying restricted "at most one". (Contributed by NM, 17-Jun-2017.) | 
| Ref | Expression | 
|---|---|
| rmo2.1 | ⊢ Ⅎ𝑦𝜑 | 
| Ref | Expression | 
|---|---|
| rmo2i | ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rexex 3075 | . 2 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) | |
| 2 | rmo2.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
| 3 | 2 | rmo2 3886 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) | 
| 4 | 1, 3 | sylibr 234 | 1 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∃wex 1778 Ⅎwnf 1782 ∀wral 3060 ∃wrex 3069 ∃*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-10 2140 ax-11 2156 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-mo 2539 df-ral 3061 df-rex 3070 df-rmo 3379 | 
| This theorem is referenced by: mndmolinv 42097 modelaxreplem2 45001 | 
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