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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 4392 | . . 3 ⊢ ⦋𝐶 / 𝑥⦌({𝐴} ∪ {𝐵}) = (⦋𝐶 / 𝑥⦌{𝐴} ∪ ⦋𝐶 / 𝑥⦌{𝐵}) | |
2 | csbsng 4646 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐴} = {⦋𝐶 / 𝑥⦌𝐴}) | |
3 | csbsng 4646 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐵} = {⦋𝐶 / 𝑥⦌𝐵}) | |
4 | 2, 3 | uneq12d 4142 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (⦋𝐶 / 𝑥⦌{𝐴} ∪ ⦋𝐶 / 𝑥⦌{𝐵}) = ({⦋𝐶 / 𝑥⦌𝐴} ∪ {⦋𝐶 / 𝑥⦌𝐵})) |
5 | 1, 4 | syl5eq 2870 | . 2 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌({𝐴} ∪ {𝐵}) = ({⦋𝐶 / 𝑥⦌𝐴} ∪ {⦋𝐶 / 𝑥⦌𝐵})) |
6 | df-pr 4572 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
7 | 6 | csbeq2i 3893 | . 2 ⊢ ⦋𝐶 / 𝑥⦌{𝐴, 𝐵} = ⦋𝐶 / 𝑥⦌({𝐴} ∪ {𝐵}) |
8 | df-pr 4572 | . 2 ⊢ {⦋𝐶 / 𝑥⦌𝐴, ⦋𝐶 / 𝑥⦌𝐵} = ({⦋𝐶 / 𝑥⦌𝐴} ∪ {⦋𝐶 / 𝑥⦌𝐵}) | |
9 | 5, 7, 8 | 3eqtr4g 2883 | 1 ⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐴, 𝐵} = {⦋𝐶 / 𝑥⦌𝐴, ⦋𝐶 / 𝑥⦌𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⦋csb 3885 ∪ cun 3936 {csn 4569 {cpr 4571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-nul 4294 df-sn 4570 df-pr 4572 |
This theorem is referenced by: csbopg 4823 |
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