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Theorem csbprg 4675
Description: Distribute proper substitution through a pair of classes. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
Assertion
Ref Expression
csbprg (𝐶𝑉𝐶 / 𝑥{𝐴, 𝐵} = {𝐶 / 𝑥𝐴, 𝐶 / 𝑥𝐵})

Proof of Theorem csbprg
StepHypRef Expression
1 csbun 4403 . . 3 𝐶 / 𝑥({𝐴} ∪ {𝐵}) = (𝐶 / 𝑥{𝐴} ∪ 𝐶 / 𝑥{𝐵})
2 csbsng 4674 . . . 4 (𝐶𝑉𝐶 / 𝑥{𝐴} = {𝐶 / 𝑥𝐴})
3 csbsng 4674 . . . 4 (𝐶𝑉𝐶 / 𝑥{𝐵} = {𝐶 / 𝑥𝐵})
42, 3uneq12d 4129 . . 3 (𝐶𝑉 → (𝐶 / 𝑥{𝐴} ∪ 𝐶 / 𝑥{𝐵}) = ({𝐶 / 𝑥𝐴} ∪ {𝐶 / 𝑥𝐵}))
51, 4eqtrid 2789 . 2 (𝐶𝑉𝐶 / 𝑥({𝐴} ∪ {𝐵}) = ({𝐶 / 𝑥𝐴} ∪ {𝐶 / 𝑥𝐵}))
6 df-pr 4594 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
76csbeq2i 3868 . 2 𝐶 / 𝑥{𝐴, 𝐵} = 𝐶 / 𝑥({𝐴} ∪ {𝐵})
8 df-pr 4594 . 2 {𝐶 / 𝑥𝐴, 𝐶 / 𝑥𝐵} = ({𝐶 / 𝑥𝐴} ∪ {𝐶 / 𝑥𝐵})
95, 7, 83eqtr4g 2802 1 (𝐶𝑉𝐶 / 𝑥{𝐴, 𝐵} = {𝐶 / 𝑥𝐴, 𝐶 / 𝑥𝐵})
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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