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| Mirrors > Home > MPE Home > Th. List > sb8euv | Structured version Visualization version GIF version | ||
| Description: Variable substitution in unique existential quantifier. Version of sb8eu 2600 requiring more disjoint variables, but fewer axioms. (Contributed by NM, 7-Aug-1994.) (Revised by Wolf Lammen, 7-Feb-2023.) | 
| Ref | Expression | 
|---|---|
| sb8euv.nf | ⊢ Ⅎ𝑦𝜑 | 
| Ref | Expression | 
|---|---|
| sb8euv | ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sb8euv.nf | . . 3 ⊢ Ⅎ𝑦𝜑 | |
| 2 | 1 | nfsbv 2330 | . 2 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑 | 
| 3 | 2 | sb8eulem 2598 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 Ⅎwnf 1783 [wsb 2064 ∃!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 ax-10 2141 ax-11 2157 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 | 
| This theorem is referenced by: eu1 2610 cbvreuwOLD 3415 | 
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