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| Mirrors > Home > MPE Home > Th. List > Mathboxes > unqsym1 | Structured version Visualization version GIF version | ||
| Description: A symmetry with ∃!. See negsym1 36418 for more information. (Contributed by Anthony Hart, 6-Sep-2011.) | 
| Ref | Expression | 
|---|---|
| unqsym1 | ⊢ (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | neufal 36407 | . . . 4 ⊢ ¬ ∃!𝑥⊥ | |
| 2 | 1 | nex 1800 | . . 3 ⊢ ¬ ∃𝑥∃!𝑥⊥ | 
| 3 | euex 2577 | . . 3 ⊢ (∃!𝑥∃!𝑥⊥ → ∃𝑥∃!𝑥⊥) | |
| 4 | 2, 3 | mto 197 | . 2 ⊢ ¬ ∃!𝑥∃!𝑥⊥ | 
| 5 | 4 | pm2.21i 119 | 1 ⊢ (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ⊥wfal 1552 ∃wex 1779 ∃!weu 2568 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-eu 2569 | 
| This theorem is referenced by: (None) | 
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