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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 2598 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 2323 | . 2 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑 |
3 | 2 | sb8eulem 2596 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 Ⅎwnf 1785 [wsb 2067 ∃!weu 2566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 |
This theorem is referenced by: cbvmowOLD 2602 cbveuwOLD 2606 eu1 2610 cbvreuwOLD 3389 |
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