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Theorem csbprg 4519
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 4274 . . 3 𝐶 / 𝑥({𝐴} ∪ {𝐵}) = (𝐶 / 𝑥{𝐴} ∪ 𝐶 / 𝑥{𝐵})
2 csbsng 4518 . . . 4 (𝐶𝑉𝐶 / 𝑥{𝐴} = {𝐶 / 𝑥𝐴})
3 csbsng 4518 . . . 4 (𝐶𝑉𝐶 / 𝑥{𝐵} = {𝐶 / 𝑥𝐵})
42, 3uneq12d 4029 . . 3 (𝐶𝑉 → (𝐶 / 𝑥{𝐴} ∪ 𝐶 / 𝑥{𝐵}) = ({𝐶 / 𝑥𝐴} ∪ {𝐶 / 𝑥𝐵}))
51, 4syl5eq 2826 . 2 (𝐶𝑉𝐶 / 𝑥({𝐴} ∪ {𝐵}) = ({𝐶 / 𝑥𝐴} ∪ {𝐶 / 𝑥𝐵}))
6 df-pr 4444 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
76csbeq2i 4257 . 2 𝐶 / 𝑥{𝐴, 𝐵} = 𝐶 / 𝑥({𝐴} ∪ {𝐵})
8 df-pr 4444 . 2 {𝐶 / 𝑥𝐴, 𝐶 / 𝑥𝐵} = ({𝐶 / 𝑥𝐴} ∪ {𝐶 / 𝑥𝐵})
95, 7, 83eqtr4g 2839 1 (𝐶𝑉𝐶 / 𝑥{𝐴, 𝐵} = {𝐶 / 𝑥𝐴, 𝐶 / 𝑥𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  csb 3786  cun 3827  {csn 4441  {cpr 4443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-nul 4179  df-sn 4442  df-pr 4444
This theorem is referenced by:  csbopg  4695
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