ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  csbunig GIF version

Theorem csbunig 3751
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 3065 . . 3 (𝐴𝑉𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)})
2 sbcexg 2966 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵)))
3 sbcang 2955 . . . . . . 7 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵)))
4 sbcg 2981 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑦𝑧𝑦))
5 sbcel2g 3027 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵))
64, 5anbi12d 465 . . . . . . 7 (𝐴𝑉 → (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
73, 6bitrd 187 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
87exbidv 1798 . . . . 5 (𝐴𝑉 → (∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
92, 8bitrd 187 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
109abbidv 2258 . . 3 (𝐴𝑉 → {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)})
111, 10eqtrd 2173 . 2 (𝐴𝑉𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)})
12 df-uni 3744 . . 3 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
1312csbeq2i 3033 . 2 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
14 df-uni 3744 . 2 𝐴 / 𝑥𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}
1511, 13, 143eqtr4g 2198 1 (𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wex 1469  wcel 1481  {cab 2126  [wsbc 2912  csb 3006   cuni 3743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2913  df-csb 3007  df-uni 3744
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator