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Theorem csbuni 4870
Description: Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 22-Aug-2018.)
Assertion
Ref Expression
csbuni 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵

Proof of Theorem csbuni
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbab 4371 . . . 4 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)}
2 sbcex2 3781 . . . . . 6 ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵))
3 sbcan 3768 . . . . . . . 8 ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵))
4 sbcg 3795 . . . . . . . . . 10 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑧𝑦𝑧𝑦))
54anbi1d 630 . . . . . . . . 9 (𝐴 ∈ V → (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦[𝐴 / 𝑥]𝑦𝐵)))
6 sbcel2 4349 . . . . . . . . . 10 ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵)
76anbi2i 623 . . . . . . . . 9 ((𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))
85, 7bitrdi 287 . . . . . . . 8 (𝐴 ∈ V → (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
93, 8bitrid 282 . . . . . . 7 (𝐴 ∈ V → ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
109exbidv 1924 . . . . . 6 (𝐴 ∈ V → (∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
112, 10bitrid 282 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
1211abbidv 2807 . . . 4 (𝐴 ∈ V → {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)})
131, 12eqtrid 2790 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)})
14 df-uni 4840 . . . 4 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
1514csbeq2i 3840 . . 3 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
16 df-uni 4840 . . 3 𝐴 / 𝑥𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}
1713, 15, 163eqtr4g 2803 . 2 (𝐴 ∈ V → 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
18 csbprc 4340 . . 3 𝐴 ∈ V → 𝐴 / 𝑥 𝐵 = ∅)
19 csbprc 4340 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
2019unieqd 4853 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
21 uni0 4869 . . . 4 ∅ = ∅
2220, 21eqtr2di 2795 . . 3 𝐴 ∈ V → ∅ = 𝐴 / 𝑥𝐵)
2318, 22eqtrd 2778 . 2 𝐴 ∈ V → 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
2417, 23pm2.61i 182 1 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1539  wex 1782  wcel 2106  {cab 2715  Vcvv 3432  [wsbc 3716  csb 3832  c0 4256   cuni 4839
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-uni 4840
This theorem is referenced by:  csbfrecsg  8100  csbfv12gALTVD  42519
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