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Mirrors > Home > MPE Home > Th. List > sb8mo | Structured version Visualization version GIF version |
Description: Variable substitution for the at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Ref | Expression |
---|---|
sb8eu.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
sb8mo | ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb8eu.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | sb8e 2553 | . . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
3 | 1 | sb8eu 2679 | . . 3 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) |
4 | 2, 3 | imbi12i 352 | . 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑)) |
5 | moeu 2661 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
6 | moeu 2661 | . 2 ⊢ (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑)) | |
7 | 4, 5, 6 | 3bitr4i 304 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∃wex 1771 Ⅎwnf 1775 [wsb 2060 ∃*wmo 2613 ∃!weu 2646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 |
This theorem is referenced by: cbvmo 2682 |
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