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Mirrors > Home > ILE Home > Th. List > iotacl | GIF version |
Description: Membership law for
descriptions.
This can useful for expanding an unbounded iota-based definition (see df-iota 5160). (Contributed by Andrew Salmon, 1-Aug-2011.) |
Ref | Expression |
---|---|
iotacl | ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iota4 5178 | . 2 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑) | |
2 | df-sbc 2956 | . 2 ⊢ ([(℩𝑥𝜑) / 𝑥]𝜑 ↔ (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | sylib 121 | 1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃!weu 2019 ∈ wcel 2141 {cab 2156 [wsbc 2955 ℩cio 5158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-sn 3589 df-pr 3590 df-uni 3797 df-iota 5160 |
This theorem is referenced by: riotacl2 5822 eroprf 6606 |
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