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| Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) | 
| Ref | Expression | 
|---|---|
| absneu | ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃!𝑥𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sneq 4635 | . . . . 5 ⊢ (𝑦 = 𝐴 → {𝑦} = {𝐴}) | |
| 2 | 1 | eqeq2d 2747 | . . . 4 ⊢ (𝑦 = 𝐴 → ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝐴})) | 
| 3 | 2 | spcegv 3596 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∣ 𝜑} = {𝐴} → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})) | 
| 4 | 3 | imp 406 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | 
| 5 | euabsn2 4724 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
| 6 | 4, 5 | sylibr 234 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃!𝑥𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ∃!weu 2567 {cab 2713 {csn 4625 | 
| 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-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-sn 4626 | 
| This theorem is referenced by: rabsneu 4728 | 
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