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

Theorem csbing 3196
Description: Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)
Assertion
Ref Expression
csbing (𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))

Proof of Theorem csbing
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 2925 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥(𝐶𝐷))
2 csbeq1 2925 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐶)
3 csbeq1 2925 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐷 = 𝐴 / 𝑥𝐷)
42, 3ineq12d 3191 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
51, 4eqeq12d 2099 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐶𝐷) = (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷) ↔ 𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)))
6 vex 2618 . . 3 𝑦 ∈ V
7 nfcsb1v 2952 . . . 4 𝑥𝑦 / 𝑥𝐶
8 nfcsb1v 2952 . . . 4 𝑥𝑦 / 𝑥𝐷
97, 8nfin 3195 . . 3 𝑥(𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷)
10 csbeq1a 2930 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
11 csbeq1a 2930 . . . 4 (𝑥 = 𝑦𝐷 = 𝑦 / 𝑥𝐷)
1210, 11ineq12d 3191 . . 3 (𝑥 = 𝑦 → (𝐶𝐷) = (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷))
136, 9, 12csbief 2961 . 2 𝑦 / 𝑥(𝐶𝐷) = (𝑦 / 𝑥𝐶𝑦 / 𝑥𝐷)
145, 13vtoclg 2672 1 (𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wcel 1436  csb 2922  cin 2987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-in 2994
This theorem is referenced by:  csbresg  4683
  Copyright terms: Public domain W3C validator