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Mirrors > Home > ILE Home > Th. List > sb8mo | 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 1734 | . . 3 ⊢ (∃xφ ↔ ∃y[y / x]φ) |
3 | 1 | sb8eu 1910 | . . 3 ⊢ (∃!xφ ↔ ∃!y[y / x]φ) |
4 | 2, 3 | imbi12i 228 | . 2 ⊢ ((∃xφ → ∃!xφ) ↔ (∃y[y / x]φ → ∃!y[y / x]φ)) |
5 | df-mo 1901 | . 2 ⊢ (∃*xφ ↔ (∃xφ → ∃!xφ)) | |
6 | df-mo 1901 | . 2 ⊢ (∃*y[y / x]φ ↔ (∃y[y / x]φ → ∃!y[y / x]φ)) | |
7 | 4, 5, 6 | 3bitr4i 201 | 1 ⊢ (∃*xφ ↔ ∃*y[y / x]φ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 Ⅎwnf 1346 ∃wex 1378 [wsb 1642 ∃!weu 1897 ∃*wmo 1898 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 |
This theorem is referenced by: (None) |
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