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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 2647 | . 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) | |
2 | eu4.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | mo4 2643 | . . 3 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) |
4 | 3 | anbi2i 622 | . 2 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))) |
5 | 1, 4 | bitri 276 | 1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 ∃wex 1771 ∃*wmo 2613 ∃!weu 2646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-mo 2615 df-eu 2647 |
This theorem is referenced by: euind 3712 eqeuel 4319 uniintsn 4904 eusv1 5282 omeu 8200 eroveu 8381 climeu 14900 pceu 16171 initoeu2lem2 17263 psgneu 18563 gsumval3eu 18953 frgr3vlem2 27980 3vfriswmgrlem 27983 unirep 34869 rlimdmafv 43253 rlimdmafv2 43334 |
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