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Theorem csbsng 4190
 Description: Distribute proper substitution through the singleton of a class. csbsng 4190 is derived from the virtual deduction proof csbsngVD 38151. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbsng (𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})

Proof of Theorem csbsng
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbab 3960 . . 3 𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦[𝐴 / 𝑥]𝑦 = 𝐵}
2 sbceq2g 3942 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵))
32abbidv 2728 . . 3 (𝐴𝑉 → {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵})
41, 3syl5eq 2656 . 2 (𝐴𝑉𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵})
5 df-sn 4126 . . 3 {𝐵} = {𝑦𝑦 = 𝐵}
65csbeq2i 3945 . 2 𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}
7 df-sn 4126 . 2 {𝐴 / 𝑥𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}
84, 6, 73eqtr4g 2669 1 (𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {cab 2596  [wsbc 3402  ⦋csb 3499  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-nul 3875  df-sn 4126 This theorem is referenced by:  csbprg  4191  csbopg  4358  csbpredg  32348  csbfv12gALTOLD  38074  csbfv12gALTVD  38157
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