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Theorem csbuni 4941
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 4438 . . . 4 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)}
2 sbcex2 3843 . . . . . 6 ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵))
3 sbcan 3830 . . . . . . . 8 ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵))
4 sbcg 3857 . . . . . . . . . 10 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑧𝑦𝑧𝑦))
54anbi1d 628 . . . . . . . . 9 (𝐴 ∈ V → (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦[𝐴 / 𝑥]𝑦𝐵)))
6 sbcel2 4416 . . . . . . . . . 10 ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵)
76anbi2i 621 . . . . . . . . 9 ((𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))
85, 7bitrdi 286 . . . . . . . 8 (𝐴 ∈ V → (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
93, 8bitrid 282 . . . . . . 7 (𝐴 ∈ V → ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
109exbidv 1922 . . . . . 6 (𝐴 ∈ V → (∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
112, 10bitrid 282 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
1211abbidv 2799 . . . 4 (𝐴 ∈ V → {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)})
131, 12eqtrid 2782 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)})
14 df-uni 4910 . . . 4 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
1514csbeq2i 3902 . . 3 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
16 df-uni 4910 . . 3 𝐴 / 𝑥𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}
1713, 15, 163eqtr4g 2795 . 2 (𝐴 ∈ V → 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
18 csbprc 4407 . . 3 𝐴 ∈ V → 𝐴 / 𝑥 𝐵 = ∅)
19 csbprc 4407 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
2019unieqd 4923 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
21 uni0 4940 . . . 4 ∅ = ∅
2220, 21eqtr2di 2787 . . 3 𝐴 ∈ V → ∅ = 𝐴 / 𝑥𝐵)
2318, 22eqtrd 2770 . 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 394   = wceq 1539  wex 1779  wcel 2104  {cab 2707  Vcvv 3472  [wsbc 3778  csb 3894  c0 4323   cuni 4909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-uni 4910
This theorem is referenced by:  csbfrecsg  8273  csbfv12gALTVD  43964
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