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Theorem csbprg 4607
 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 4348 . . 3 𝐶 / 𝑥({𝐴} ∪ {𝐵}) = (𝐶 / 𝑥{𝐴} ∪ 𝐶 / 𝑥{𝐵})
2 csbsng 4606 . . . 4 (𝐶𝑉𝐶 / 𝑥{𝐴} = {𝐶 / 𝑥𝐴})
3 csbsng 4606 . . . 4 (𝐶𝑉𝐶 / 𝑥{𝐵} = {𝐶 / 𝑥𝐵})
42, 3uneq12d 4093 . . 3 (𝐶𝑉 → (𝐶 / 𝑥{𝐴} ∪ 𝐶 / 𝑥{𝐵}) = ({𝐶 / 𝑥𝐴} ∪ {𝐶 / 𝑥𝐵}))
51, 4syl5eq 2845 . 2 (𝐶𝑉𝐶 / 𝑥({𝐴} ∪ {𝐵}) = ({𝐶 / 𝑥𝐴} ∪ {𝐶 / 𝑥𝐵}))
6 df-pr 4530 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
76csbeq2i 3837 . 2 𝐶 / 𝑥{𝐴, 𝐵} = 𝐶 / 𝑥({𝐴} ∪ {𝐵})
8 df-pr 4530 . 2 {𝐶 / 𝑥𝐴, 𝐶 / 𝑥𝐵} = ({𝐶 / 𝑥𝐴} ∪ {𝐶 / 𝑥𝐵})
95, 7, 83eqtr4g 2858 1 (𝐶𝑉𝐶 / 𝑥{𝐴, 𝐵} = {𝐶 / 𝑥𝐴, 𝐶 / 𝑥𝐵})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ⦋csb 3829   ∪ cun 3880  {csn 4527  {cpr 4529 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-nul 4246  df-sn 4528  df-pr 4530 This theorem is referenced by:  csbopg  4786
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