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Mirrors > Home > ILE Home > Th. List > riotasbc | GIF version |
Description: Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
riotasbc | ⊢ (∃!x ∈ A φ → [(℩x ∈ A φ) / x]φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabssab 3021 | . . 3 ⊢ {x ∈ A ∣ φ} ⊆ {x ∣ φ} | |
2 | riotacl2 5424 | . . 3 ⊢ (∃!x ∈ A φ → (℩x ∈ A φ) ∈ {x ∈ A ∣ φ}) | |
3 | 1, 2 | sseldi 2937 | . 2 ⊢ (∃!x ∈ A φ → (℩x ∈ A φ) ∈ {x ∣ φ}) |
4 | df-sbc 2759 | . 2 ⊢ ([(℩x ∈ A φ) / x]φ ↔ (℩x ∈ A φ) ∈ {x ∣ φ}) | |
5 | 3, 4 | sylibr 137 | 1 ⊢ (∃!x ∈ A φ → [(℩x ∈ A φ) / x]φ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1390 {cab 2023 ∃!wreu 2302 {crab 2304 [wsbc 2758 ℩crio 5410 |
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 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-uni 3572 df-iota 4810 df-riota 5411 |
This theorem is referenced by: riotass2 5437 riotass 5438 cjth 9074 |
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