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Mirrors > Home > MPE Home > Th. List > sb8eu | Structured version Visualization version GIF version |
Description: Variable substitution in unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2386. For a version requiring more disjoint variables, but fewer axioms, see sb8euv 2681. (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 2561 | . 2 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑 |
3 | 2 | sb8eulem 2680 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 Ⅎwnf 1780 [wsb 2065 ∃!weu 2649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-11 2156 ax-12 2172 ax-13 2386 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 |
This theorem is referenced by: sb8mo 2683 cbveu 2687 cbvreu 3448 |
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