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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 4434 is derived from the virtual deduction proof csbsngVD 39884. (Contributed by Alan Sare, 10-Nov-2012.) |
Ref | Expression |
---|---|
csbsng | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbab 4205 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} | |
2 | sbceq2g 4186 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)) | |
3 | 2 | abbidv 2919 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}) |
4 | 1, 3 | syl5eq 2846 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}) |
5 | df-sn 4370 | . . 3 ⊢ {𝐵} = {𝑦 ∣ 𝑦 = 𝐵} | |
6 | 5 | csbeq2i 4189 | . 2 ⊢ ⦋𝐴 / 𝑥⦌{𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} |
7 | df-sn 4370 | . 2 ⊢ {⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} | |
8 | 4, 6, 7 | 3eqtr4g 2859 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 {cab 2786 [wsbc 3634 ⦋csb 3729 {csn 4369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-nul 4117 df-sn 4370 |
This theorem is referenced by: csbprg 4435 csbopg 4612 csbpredg 33670 csbfv12gALTVD 39890 |
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