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| Mirrors > Home > MPE Home > Th. List > eu4 | Structured version Visualization version GIF version | ||
| Description: Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.) | 
| Ref | Expression | 
|---|---|
| eu4.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| eu4 | ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-eu 2569 | . 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) | |
| 2 | eu4.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 3 | 2 | mo4 2566 | . . 3 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) | 
| 4 | 3 | anbi2i 623 | . 2 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))) | 
| 5 | 1, 4 | bitri 275 | 1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 ∃wex 1779 ∃*wmo 2538 ∃!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-ex 1780 df-mo 2540 df-eu 2569 | 
| This theorem is referenced by: euind 3730 eqeuel 4365 uniintsn 4985 eusv1 5391 omeu 8623 eroveu 8852 climeu 15591 pceu 16884 initoeu2lem2 18060 psgneu 19524 gsumval3eu 19922 frgr3vlem2 30293 3vfriswmgrlem 30296 unirep 37721 rlimdmafv 47189 rlimdmafv2 47270 | 
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