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Mirrors > Home > MPE Home > Th. List > csbsng | Structured version Visualization version GIF version |
Description: Distribute proper substitution through the singleton of a class. csbsng 4708 is derived from the virtual deduction proof csbsngVD 44255. (Contributed by Alan Sare, 10-Nov-2012.) |
Ref | Expression |
---|---|
csbsng | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbab 4433 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} | |
2 | sbceq2g 4412 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)) | |
3 | 2 | abbidv 2796 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}) |
4 | 1, 3 | eqtrid 2779 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}) |
5 | df-sn 4625 | . . 3 ⊢ {𝐵} = {𝑦 ∣ 𝑦 = 𝐵} | |
6 | 5 | csbeq2i 3897 | . 2 ⊢ ⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} |
7 | df-sn 4625 | . 2 ⊢ {⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} | |
8 | 4, 6, 7 | 3eqtr4g 2792 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {cab 2704 [wsbc 3774 ⦋csb 3889 {csn 4624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-nul 4319 df-sn 4625 |
This theorem is referenced by: csbprg 4709 csbopg 4887 csbpredg 6305 csbfv12gALTVD 44261 |
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