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| Description: Alternate definition of restricted "at most one". Note that ∃*𝑥 ∈ 𝐴𝜑 is not equivalent to ∃𝑦 ∈ 𝐴∀𝑥 ∈ 𝐴(𝜑 → 𝑥 = 𝑦) (in analogy to reu6 3732); to see this, let 𝐴 be the empty set. However, one direction of this pattern holds; see rmo2i 3888. (Contributed by NM, 17-Jun-2017.) | 
| Ref | Expression | 
|---|---|
| rmo2.1 | ⊢ Ⅎ𝑦𝜑 | 
| Ref | Expression | 
|---|---|
| rmo2 | ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-rmo 3380 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 2 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
| 3 | rmo2.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 4 | 2, 3 | nfan 1899 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) | 
| 5 | 4 | mof 2563 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦)) | 
| 6 | impexp 450 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦))) | |
| 7 | 6 | albii 1819 | . . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦))) | 
| 8 | df-ral 3062 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦))) | |
| 9 | 7, 8 | bitr4i 278 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) | 
| 10 | 9 | exbii 1848 | . 2 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) | 
| 11 | 1, 5, 10 | 3bitri 297 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 ∃wex 1779 Ⅎwnf 1783 ∈ wcel 2108 ∃*wmo 2538 ∀wral 3061 ∃*wrmo 3379 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-mo 2540 df-ral 3062 df-rmo 3380 | 
| This theorem is referenced by: rmo2i 3888 rmoanimALT 3895 disjiun 5131 poimirlem2 37629 onsucf1lem 43282 | 
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