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Theorem csbunig 3588
 Description: Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbunig (𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)

Proof of Theorem csbunig
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabg 2907 . . 3 (𝐴𝑉𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)})
2 sbcexg 2813 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵)))
3 sbcang 2806 . . . . . . 7 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵)))
4 sbcg 2827 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑦𝑧𝑦))
5 sbcel2g 2871 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵))
64, 5anbi12d 442 . . . . . . 7 (𝐴𝑉 → (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
73, 6bitrd 177 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
87exbidv 1706 . . . . 5 (𝐴𝑉 → (∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
92, 8bitrd 177 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
109abbidv 2155 . . 3 (𝐴𝑉 → {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)})
111, 10eqtrd 2072 . 2 (𝐴𝑉𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)})
12 df-uni 3581 . . 3 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
1312csbeq2i 2876 . 2 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
14 df-uni 3581 . 2 𝐴 / 𝑥𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}
1511, 13, 143eqtr4g 2097 1 (𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  [wsbc 2764  ⦋csb 2852  ∪ cuni 3580 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853  df-uni 3581 This theorem is referenced by: (None)
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