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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 2566 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | vex 2560 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | df-csb 2853 | . . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝑥} | |
4 | sbcel2gv 2822 | . . . . . . . 8 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | |
5 | 4 | abbi1dv 2157 | . . . . . . 7 ⊢ (𝑦 ∈ V → {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝑥} = 𝑦) |
6 | 3, 5 | syl5eq 2084 | . . . . . 6 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑥 = 𝑦) |
7 | 2, 6 | ax-mp 7 | . . . . 5 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦 |
8 | 7 | csbeq2i 2876 | . . . 4 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝑥 = ⦋𝐴 / 𝑦⦌𝑦 |
9 | csbco 2861 | . . . 4 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝑥 = ⦋𝐴 / 𝑥⦌𝑥 | |
10 | df-csb 2853 | . . . 4 ⊢ ⦋𝐴 / 𝑦⦌𝑦 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦} | |
11 | 8, 9, 10 | 3eqtr3i 2068 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝑥 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦} |
12 | sbcel2gv 2822 | . . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴)) | |
13 | 12 | abbi1dv 2157 | . . 3 ⊢ (𝐴 ∈ V → {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦} = 𝐴) |
14 | 11, 13 | syl5eq 2084 | . 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 1243 ∈ wcel 1393 {cab 2026 Vcvv 2557 [wsbc 2764 ⦋csb 2852 |
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-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 |
This theorem is referenced by: sbccsb2g 2879 csbfvg 5211 bj-sels 10034 |
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