ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ineq2 Structured version   GIF version

Theorem ineq2 3126
Description: Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
Assertion
Ref Expression
ineq2 (A = B → (𝐶A) = (𝐶B))

Proof of Theorem ineq2
StepHypRef Expression
1 ineq1 3125 . 2 (A = B → (A𝐶) = (B𝐶))
2 incom 3123 . 2 (𝐶A) = (A𝐶)
3 incom 3123 . 2 (𝐶B) = (B𝐶)
41, 2, 33eqtr4g 2094 1 (A = B → (𝐶A) = (𝐶B))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  cin 2910
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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918
This theorem is referenced by:  ineq12  3127  ineq2i  3129  ineq2d  3132  uneqin  3182  intprg  3639
  Copyright terms: Public domain W3C validator