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| Description: Two ways to express "exactly one thing exists". The left-hand side requires only one variable to express this. Both sides are false in set theory, see Theorem dtru 5441. (Contributed by NM, 5-Apr-2004.) (Proof shortened by BJ, 7-Oct-2022.) | 
| Ref | Expression | 
|---|---|
| exists1 | ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | equid 2011 | . . . 4 ⊢ 𝑥 = 𝑥 | |
| 2 | 1 | bitru 1549 | . . 3 ⊢ (𝑥 = 𝑥 ↔ ⊤) | 
| 3 | 2 | eubii 2585 | . 2 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∃!𝑥⊤) | 
| 4 | euae 2660 | . 2 ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) | |
| 5 | 3, 4 | bitri 275 | 1 ⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∀wal 1538 ⊤wtru 1541 ∃!weu 2568 | 
| 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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-mo 2540 df-eu 2569 | 
| This theorem is referenced by: exists2 2662 | 
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