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Mirrors > Home > ILE Home > Th. List > sbc8g | GIF version |
Description: This is the closest we can get to df-sbc 2759 if we start from dfsbcq 2760 (see its comments) and dfsbcq2 2761. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
sbc8g | ⊢ (A ∈ 𝑉 → ([A / x]φ ↔ A ∈ {x ∣ φ})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 2760 | . 2 ⊢ (y = A → ([y / x]φ ↔ [A / x]φ)) | |
2 | eleq1 2097 | . 2 ⊢ (y = A → (y ∈ {x ∣ φ} ↔ A ∈ {x ∣ φ})) | |
3 | df-clab 2024 | . . 3 ⊢ (y ∈ {x ∣ φ} ↔ [y / x]φ) | |
4 | equid 1586 | . . . 4 ⊢ y = y | |
5 | dfsbcq2 2761 | . . . 4 ⊢ (y = y → ([y / x]φ ↔ [y / x]φ)) | |
6 | 4, 5 | ax-mp 7 | . . 3 ⊢ ([y / x]φ ↔ [y / x]φ) |
7 | 3, 6 | bitr2i 174 | . 2 ⊢ ([y / x]φ ↔ y ∈ {x ∣ φ}) |
8 | 1, 2, 7 | vtoclbg 2608 | 1 ⊢ (A ∈ 𝑉 → ([A / x]φ ↔ A ∈ {x ∣ φ})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∈ wcel 1390 [wsb 1642 {cab 2023 [wsbc 2758 |
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-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-sbc 2759 |
This theorem is referenced by: bj-elssuniab 9265 |
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