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Theorem csbdm 5240
 Description: Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.) (Revised by NM, 24-Aug-2018.)
Assertion
Ref Expression
csbdm 𝐴 / 𝑥dom 𝐵 = dom 𝐴 / 𝑥𝐵

Proof of Theorem csbdm
Dummy variables 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbab 3960 . . 3 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦[𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵}
2 sbcex2 3453 . . . . 5 ([𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]𝑦, 𝑤⟩ ∈ 𝐵)
3 sbcel2 3941 . . . . . 6 ([𝐴 / 𝑥]𝑦, 𝑤⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵)
43exbii 1764 . . . . 5 (∃𝑤[𝐴 / 𝑥]𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵)
52, 4bitri 263 . . . 4 ([𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵)
65abbii 2726 . . 3 {𝑦[𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵}
71, 6eqtri 2632 . 2 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵}
8 dfdm3 5232 . . 3 dom 𝐵 = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵}
98csbeq2i 3945 . 2 𝐴 / 𝑥dom 𝐵 = 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵}
10 dfdm3 5232 . 2 dom 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵}
117, 9, 103eqtr4i 2642 1 𝐴 / 𝑥dom 𝐵 = dom 𝐴 / 𝑥𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  [wsbc 3402  ⦋csb 3499  ⟨cop 4131  dom cdm 5038 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-br 4584  df-dm 5048 This theorem is referenced by:  sbcfng  5955
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