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Theorem csbdm 5854
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 4398 . . 3 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦[𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵}
2 sbcex2 3805 . . . . 5 ([𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]𝑦, 𝑤⟩ ∈ 𝐵)
3 sbcel2 4376 . . . . . 6 ([𝐴 / 𝑥]𝑦, 𝑤⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵)
43exbii 1851 . . . . 5 (∃𝑤[𝐴 / 𝑥]𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵)
52, 4bitri 275 . . . 4 ([𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵)
65abbii 2803 . . 3 {𝑦[𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵}
71, 6eqtri 2761 . 2 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵}
8 dfdm3 5844 . . 3 dom 𝐵 = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵}
98csbeq2i 3864 . 2 𝐴 / 𝑥dom 𝐵 = 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵}
10 dfdm3 5844 . 2 dom 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵}
117, 9, 103eqtr4i 2771 1 𝐴 / 𝑥dom 𝐵 = dom 𝐴 / 𝑥𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wex 1782  wcel 2107  {cab 2710  [wsbc 3740  csb 3856  cop 4593  dom cdm 5634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-nul 4284  df-br 5107  df-dm 5644
This theorem is referenced by:  sbcfng  6666
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