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Mirrors > Home > ILE Home > Th. List > csbopabg | GIF version |
Description: Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) |
Ref | Expression |
---|---|
csbopabg | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 2855 | . . 3 ⊢ (𝑤 = 𝐴 → ⦋𝑤 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑}) | |
2 | dfsbcq2 2767 | . . . 4 ⊢ (𝑤 = 𝐴 → ([𝑤 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | opabbidv 3823 | . . 3 ⊢ (𝑤 = 𝐴 → {〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝜑}) |
4 | 1, 3 | eqeq12d 2054 | . 2 ⊢ (𝑤 = 𝐴 → (⦋𝑤 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑} ↔ ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝜑})) |
5 | vex 2560 | . . 3 ⊢ 𝑤 ∈ V | |
6 | nfs1v 1815 | . . . 4 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑 | |
7 | 6 | nfopab 3825 | . . 3 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑} |
8 | sbequ12 1654 | . . . 4 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑)) | |
9 | 8 | opabbidv 3823 | . . 3 ⊢ (𝑥 = 𝑤 → {〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑}) |
10 | 5, 7, 9 | csbief 2891 | . 2 ⊢ ⦋𝑤 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝑤 / 𝑥]𝜑} |
11 | 4, 10 | vtoclg 2613 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝜑}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 [wsb 1645 [wsbc 2764 ⦋csb 2852 {copab 3817 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-sbc 2765 df-csb 2853 df-opab 3819 |
This theorem is referenced by: csbcnvg 4519 |
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