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| Description: Variable substitution in unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2377. For a version requiring more disjoint variables, but fewer axioms, see sb8euv 2599. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| sb8eu.1 | ⊢ Ⅎ𝑦𝜑 | 
| Ref | Expression | 
|---|---|
| sb8eu | ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sb8eu.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
| 2 | 1 | nfsb 2528 | . 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 ax-13 2377 | 
| 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: sb8mo 2601 cbveu 2607 cbvreu 3428 | 
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