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Theorem csbdm 5473
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 4151 . . 3 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦[𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵}
2 sbcex2 3627 . . . . 5 ([𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]𝑦, 𝑤⟩ ∈ 𝐵)
3 sbcel2 4132 . . . . . 6 ([𝐴 / 𝑥]𝑦, 𝑤⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵)
43exbii 1923 . . . . 5 (∃𝑤[𝐴 / 𝑥]𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵)
52, 4bitri 264 . . . 4 ([𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵)
65abbii 2877 . . 3 {𝑦[𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵}
71, 6eqtri 2782 . 2 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵}
8 dfdm3 5465 . . 3 dom 𝐵 = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵}
98csbeq2i 4136 . 2 𝐴 / 𝑥dom 𝐵 = 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵}
10 dfdm3 5465 . 2 dom 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵}
117, 9, 103eqtr4i 2792 1 𝐴 / 𝑥dom 𝐵 = dom 𝐴 / 𝑥𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wex 1853  wcel 2139  {cab 2746  [wsbc 3576  csb 3674  cop 4327  dom cdm 5266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-nul 4059  df-br 4805  df-dm 5276
This theorem is referenced by:  sbcfng  6203
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