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Mirrors > Home > MPE Home > Th. List > sb8mo | Structured version Visualization version GIF version |
Description: Variable substitution for "at most one." (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 2413 | . . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
3 | 1 | sb8eu 2491 | . . 3 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) |
4 | 2, 3 | imbi12i 339 | . 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑)) |
5 | df-mo 2463 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
6 | df-mo 2463 | . 2 ⊢ (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑)) | |
7 | 4, 5, 6 | 3bitr4i 291 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∃wex 1695 Ⅎwnf 1699 [wsb 1867 ∃!weu 2458 ∃*wmo 2459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 |
This theorem is referenced by: (None) |
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