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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 4403 | . . 3 ⊢ ⦋𝐶 / 𝑥⦌({𝐴} ∪ {𝐵}) = (⦋𝐶 / 𝑥⦌{𝐴} ∪ ⦋𝐶 / 𝑥⦌{𝐵}) | |
2 | csbsng 4674 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐴} = {⦋𝐶 / 𝑥⦌𝐴}) | |
3 | csbsng 4674 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐵} = {⦋𝐶 / 𝑥⦌𝐵}) | |
4 | 2, 3 | uneq12d 4129 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (⦋𝐶 / 𝑥⦌{𝐴} ∪ ⦋𝐶 / 𝑥⦌{𝐵}) = ({⦋𝐶 / 𝑥⦌𝐴} ∪ {⦋𝐶 / 𝑥⦌𝐵})) |
5 | 1, 4 | eqtrid 2789 | . 2 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌({𝐴} ∪ {𝐵}) = ({⦋𝐶 / 𝑥⦌𝐴} ∪ {⦋𝐶 / 𝑥⦌𝐵})) |
6 | df-pr 4594 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
7 | 6 | csbeq2i 3868 | . 2 ⊢ ⦋𝐶 / 𝑥⦌{𝐴, 𝐵} = ⦋𝐶 / 𝑥⦌({𝐴} ∪ {𝐵}) |
8 | df-pr 4594 | . 2 ⊢ {⦋𝐶 / 𝑥⦌𝐴, ⦋𝐶 / 𝑥⦌𝐵} = ({⦋𝐶 / 𝑥⦌𝐴} ∪ {⦋𝐶 / 𝑥⦌𝐵}) | |
9 | 5, 7, 8 | 3eqtr4g 2802 | 1 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐴, 𝐵} = {⦋𝐶 / 𝑥⦌𝐴, ⦋𝐶 / 𝑥⦌𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ⦋csb 3860 ∪ cun 3913 {csn 4591 {cpr 4593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-nul 4288 df-sn 4592 df-pr 4594 |
This theorem is referenced by: csbopg 4853 |
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