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Theorem csbdmg 4557
 Description: Distribute proper substitution through the domain of a class. (Contributed by Jim Kingdon, 8-Dec-2018.)
Assertion
Ref Expression
csbdmg (𝐴𝑉𝐴 / 𝑥dom 𝐵 = dom 𝐴 / 𝑥𝐵)

Proof of Theorem csbdmg
Dummy variables 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabg 2935 . . 3 (𝐴𝑉𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦[𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵})
2 sbcex2 2839 . . . . 5 ([𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]𝑦, 𝑤⟩ ∈ 𝐵)
3 sbcel2g 2899 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦, 𝑤⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵))
43exbidv 1722 . . . . 5 (𝐴𝑉 → (∃𝑤[𝐴 / 𝑥]𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵))
52, 4syl5bb 185 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵))
65abbidv 2171 . . 3 (𝐴𝑉 → {𝑦[𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵})
71, 6eqtrd 2088 . 2 (𝐴𝑉𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵})
8 dfdm3 4550 . . 3 dom 𝐵 = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵}
98csbeq2i 2904 . 2 𝐴 / 𝑥dom 𝐵 = 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵}
10 dfdm3 4550 . 2 dom 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵}
117, 9, 103eqtr4g 2113 1 (𝐴𝑉𝐴 / 𝑥dom 𝐵 = dom 𝐴 / 𝑥𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259  ∃wex 1397   ∈ wcel 1409  {cab 2042  [wsbc 2787  ⦋csb 2880  ⟨cop 3406  dom cdm 4373 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2788  df-csb 2881  df-br 3793  df-dm 4383 This theorem is referenced by:  sbcfng  5072
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