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Mirrors > Home > ILE Home > Th. List > csbvarg | GIF version |
Description: The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.) |
Ref | Expression |
---|---|
csbvarg | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2610 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | vex 2604 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | df-csb 2909 | . . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝑥} | |
4 | sbcel2gv 2877 | . . . . . . . 8 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | |
5 | 4 | abbi1dv 2198 | . . . . . . 7 ⊢ (𝑦 ∈ V → {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝑥} = 𝑦) |
6 | 3, 5 | syl5eq 2125 | . . . . . 6 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑥 = 𝑦) |
7 | 2, 6 | ax-mp 7 | . . . . 5 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦 |
8 | 7 | csbeq2i 2932 | . . . 4 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝑥 = ⦋𝐴 / 𝑦⦌𝑦 |
9 | csbco 2917 | . . . 4 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝑥 = ⦋𝐴 / 𝑥⦌𝑥 | |
10 | df-csb 2909 | . . . 4 ⊢ ⦋𝐴 / 𝑦⦌𝑦 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦} | |
11 | 8, 9, 10 | 3eqtr3i 2109 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝑥 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦} |
12 | sbcel2gv 2877 | . . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴)) | |
13 | 12 | abbi1dv 2198 | . . 3 ⊢ (𝐴 ∈ V → {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦} = 𝐴) |
14 | 11, 13 | syl5eq 2125 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴) |
15 | 1, 14 | syl 14 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∈ wcel 1433 {cab 2067 Vcvv 2601 [wsbc 2815 ⦋csb 2908 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sbc 2816 df-csb 2909 |
This theorem is referenced by: sbccsb2g 2935 csbfvg 5232 f1od2 5876 bj-sels 10705 |
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