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Mirrors > Home > MPE Home > Th. List > csbprg | Structured version Visualization version GIF version |
Description: Distribute proper substitution through a pair of classes. (Contributed by Alexander van der Vekens, 4-Sep-2018.) |
Ref | Expression |
---|---|
csbprg | ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐴, 𝐵} = {⦋𝐶 / 𝑥⦌𝐴, ⦋𝐶 / 𝑥⦌𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbun 4436 | . . 3 ⊢ ⦋𝐶 / 𝑥⦌({𝐴} ∪ {𝐵}) = (⦋𝐶 / 𝑥⦌{𝐴} ∪ ⦋𝐶 / 𝑥⦌{𝐵}) | |
2 | csbsng 4708 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐴} = {⦋𝐶 / 𝑥⦌𝐴}) | |
3 | csbsng 4708 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐵} = {⦋𝐶 / 𝑥⦌𝐵}) | |
4 | 2, 3 | uneq12d 4162 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (⦋𝐶 / 𝑥⦌{𝐴} ∪ ⦋𝐶 / 𝑥⦌{𝐵}) = ({⦋𝐶 / 𝑥⦌𝐴} ∪ {⦋𝐶 / 𝑥⦌𝐵})) |
5 | 1, 4 | eqtrid 2778 | . 2 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌({𝐴} ∪ {𝐵}) = ({⦋𝐶 / 𝑥⦌𝐴} ∪ {⦋𝐶 / 𝑥⦌𝐵})) |
6 | df-pr 4627 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
7 | 6 | csbeq2i 3900 | . 2 ⊢ ⦋𝐶 / 𝑥⦌{𝐴, 𝐵} = ⦋𝐶 / 𝑥⦌({𝐴} ∪ {𝐵}) |
8 | df-pr 4627 | . 2 ⊢ {⦋𝐶 / 𝑥⦌𝐴, ⦋𝐶 / 𝑥⦌𝐵} = ({⦋𝐶 / 𝑥⦌𝐴} ∪ {⦋𝐶 / 𝑥⦌𝐵}) | |
9 | 5, 7, 8 | 3eqtr4g 2791 | 1 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐴, 𝐵} = {⦋𝐶 / 𝑥⦌𝐴, ⦋𝐶 / 𝑥⦌𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⦋csb 3892 ∪ cun 3945 {csn 4624 {cpr 4626 |
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 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-nul 4324 df-sn 4625 df-pr 4627 |
This theorem is referenced by: csbopg 4890 |
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