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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 4447 . . 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 4179 . . 3 (𝐶𝑉 → (𝐶 / 𝑥{𝐴} ∪ 𝐶 / 𝑥{𝐵}) = ({𝐶 / 𝑥𝐴} ∪ {𝐶 / 𝑥𝐵}))
51, 4eqtrid 2787 . 2 (𝐶𝑉𝐶 / 𝑥({𝐴} ∪ {𝐵}) = ({𝐶 / 𝑥𝐴} ∪ {𝐶 / 𝑥𝐵}))
6 df-pr 4634 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
76csbeq2i 3916 . 2 𝐶 / 𝑥{𝐴, 𝐵} = 𝐶 / 𝑥({𝐴} ∪ {𝐵})
8 df-pr 4634 . 2 {𝐶 / 𝑥𝐴, 𝐶 / 𝑥𝐵} = ({𝐶 / 𝑥𝐴} ∪ {𝐶 / 𝑥𝐵})
95, 7, 83eqtr4g 2800 1 (𝐶𝑉𝐶 / 𝑥{𝐴, 𝐵} = {𝐶 / 𝑥𝐴, 𝐶 / 𝑥𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  csb 3908  cun 3961  {csn 4631  {cpr 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-nul 4340  df-sn 4632  df-pr 4634
This theorem is referenced by:  csbopg  4896
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