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Theorem csbdm 5485
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 4169 . . 3 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦[𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵}
2 sbcex2 3646 . . . . 5 ([𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]𝑦, 𝑤⟩ ∈ 𝐵)
3 sbcel2 4149 . . . . . 6 ([𝐴 / 𝑥]𝑦, 𝑤⟩ ∈ 𝐵 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵)
43exbii 1943 . . . . 5 (∃𝑤[𝐴 / 𝑥]𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵)
52, 4bitri 266 . . . 4 ([𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵 ↔ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵)
65abbii 2881 . . 3 {𝑦[𝐴 / 𝑥]𝑤𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵}
71, 6eqtri 2786 . 2 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵}
8 dfdm3 5477 . . 3 dom 𝐵 = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵}
98csbeq2i 4153 . 2 𝐴 / 𝑥dom 𝐵 = 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐵}
10 dfdm3 5477 . 2 dom 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑤𝑦, 𝑤⟩ ∈ 𝐴 / 𝑥𝐵}
117, 9, 103eqtr4i 2796 1 𝐴 / 𝑥dom 𝐵 = dom 𝐴 / 𝑥𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  wex 1874  wcel 2155  {cab 2750  [wsbc 3595  csb 3690  cop 4339  dom cdm 5276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-nul 4079  df-br 4809  df-dm 5286
This theorem is referenced by:  sbcfng  6219
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