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Theorem csbprg 4714
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 4439 . . 3 𝐶 / 𝑥({𝐴} ∪ {𝐵}) = (𝐶 / 𝑥{𝐴} ∪ 𝐶 / 𝑥{𝐵})
2 csbsng 4713 . . . 4 (𝐶𝑉𝐶 / 𝑥{𝐴} = {𝐶 / 𝑥𝐴})
3 csbsng 4713 . . . 4 (𝐶𝑉𝐶 / 𝑥{𝐵} = {𝐶 / 𝑥𝐵})
42, 3uneq12d 4163 . . 3 (𝐶𝑉 → (𝐶 / 𝑥{𝐴} ∪ 𝐶 / 𝑥{𝐵}) = ({𝐶 / 𝑥𝐴} ∪ {𝐶 / 𝑥𝐵}))
51, 4eqtrid 2780 . 2 (𝐶𝑉𝐶 / 𝑥({𝐴} ∪ {𝐵}) = ({𝐶 / 𝑥𝐴} ∪ {𝐶 / 𝑥𝐵}))
6 df-pr 4632 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
76csbeq2i 3900 . 2 𝐶 / 𝑥{𝐴, 𝐵} = 𝐶 / 𝑥({𝐴} ∪ {𝐵})
8 df-pr 4632 . 2 {𝐶 / 𝑥𝐴, 𝐶 / 𝑥𝐵} = ({𝐶 / 𝑥𝐴} ∪ {𝐶 / 𝑥𝐵})
95, 7, 83eqtr4g 2793 1 (𝐶𝑉𝐶 / 𝑥{𝐴, 𝐵} = {𝐶 / 𝑥𝐴, 𝐶 / 𝑥𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  csb 3892  cun 3945  {csn 4629  {cpr 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-nul 4324  df-sn 4630  df-pr 4632
This theorem is referenced by:  csbopg  4892
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