ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ineq1 Structured version   GIF version

Theorem ineq1 3125
Description: Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)
Assertion
Ref Expression
ineq1 (A = B → (A𝐶) = (B𝐶))

Proof of Theorem ineq1
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2098 . . . 4 (A = B → (x Ax B))
21anbi1d 438 . . 3 (A = B → ((x A x 𝐶) ↔ (x B x 𝐶)))
3 elin 3120 . . 3 (x (A𝐶) ↔ (x A x 𝐶))
4 elin 3120 . . 3 (x (B𝐶) ↔ (x B x 𝐶))
52, 3, 43bitr4g 212 . 2 (A = B → (x (A𝐶) ↔ x (B𝐶)))
65eqrdv 2035 1 (A = B → (A𝐶) = (B𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  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:  ineq2  3126  ineq12  3127  ineq1i  3128  ineq1d  3131  dfrab3ss  3209  intprg  3639  inex1g  3884  reseq1  4549  bdinex1g  9356
  Copyright terms: Public domain W3C validator