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

Theorem csbrng 4725
Description: Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbrng (A 𝑉A / xran B = ran A / xB)

Proof of Theorem csbrng
Dummy variables w y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabg 2901 . . 3 (A 𝑉A / x{yww, y B} = {y[A / x]ww, y B})
2 sbcexg 2807 . . . . 5 (A 𝑉 → ([A / x]ww, y Bw[A / x]w, y B))
3 sbcel2g 2865 . . . . . 6 (A 𝑉 → ([A / x]w, y B ↔ ⟨w, y A / xB))
43exbidv 1703 . . . . 5 (A 𝑉 → (w[A / x]w, y Bww, y A / xB))
52, 4bitrd 177 . . . 4 (A 𝑉 → ([A / x]ww, y Bww, y A / xB))
65abbidv 2152 . . 3 (A 𝑉 → {y[A / x]ww, y B} = {yww, y A / xB})
71, 6eqtrd 2069 . 2 (A 𝑉A / x{yww, y B} = {yww, y A / xB})
8 dfrn3 4467 . . 3 ran B = {yww, y B}
98csbeq2i 2870 . 2 A / xran B = A / x{yww, y B}
10 dfrn3 4467 . 2 ran A / xB = {yww, y A / xB}
117, 9, 103eqtr4g 2094 1 (A 𝑉A / xran B = ran A / xB)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  wex 1378   wcel 1390  {cab 2023  [wsbc 2758  csb 2846  cop 3370  ran crn 4289
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  sbcfg  4988
  Copyright terms: Public domain W3C validator