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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 2390. For a version requiring more disjoint variables, but fewer axioms, see sb8euv 2685. (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 2565 | . 2 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑 |
3 | 2 | sb8eulem 2684 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 Ⅎwnf 1784 [wsb 2069 ∃!weu 2653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 |
This theorem is referenced by: sb8mo 2687 cbveu 2691 cbvreu 3449 |
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