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Theorem csbunigOLD 38073
 Description: Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 22-Aug-2018. Use csbuni 4402 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbunigOLD (𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)

Proof of Theorem csbunigOLD
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabgOLD 38072 . . 3 (𝐴𝑉𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)})
2 sbcexgOLD 37774 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵)))
3 sbcangOLD 37760 . . . . . . 7 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵)))
4 sbcg 3470 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑦𝑧𝑦))
5 sbcel2gOLD 37776 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵))
64, 5anbi12d 743 . . . . . . 7 (𝐴𝑉 → (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
73, 6bitrd 267 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
87exbidv 1837 . . . . 5 (𝐴𝑉 → (∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
92, 8bitrd 267 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
109abbidv 2728 . . 3 (𝐴𝑉 → {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)})
111, 10eqtrd 2644 . 2 (𝐴𝑉𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)})
12 df-uni 4373 . . 3 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
1312csbeq2i 3945 . 2 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
14 df-uni 4373 . 2 𝐴 / 𝑥𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}
1511, 13, 143eqtr4g 2669 1 (𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  [wsbc 3402  ⦋csb 3499  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500  df-uni 4373 This theorem is referenced by: (None)
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