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| Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.) | 
| Ref | Expression | 
|---|---|
| euabsn2 | ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eu6 2573 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
| 2 | absn 4644 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
| 3 | 2 | exbii 1847 | . 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | 
| 4 | 1, 3 | bitr4i 278 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∀wal 1537 = wceq 1539 ∃wex 1778 ∃!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-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-sn 4626 | 
| This theorem is referenced by: euabsn 4725 reusn 4726 absneu 4727 uniintab 4985 eusvobj2 7424 euabsneu 47045 aiotaexb 47106 | 
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