ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ineq1d Structured version   GIF version

Theorem ineq1d 3131
Description: Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
Hypothesis
Ref Expression
ineq1d.1 (φA = B)
Assertion
Ref Expression
ineq1d (φ → (A𝐶) = (B𝐶))

Proof of Theorem ineq1d
StepHypRef Expression
1 ineq1d.1 . 2 (φA = B)
2 ineq1 3125 . 2 (A = B → (A𝐶) = (B𝐶))
31, 2syl 14 1 (φ → (A𝐶) = (B𝐶))
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:  diftpsn3  3496  ordpwsucexmid  4246  riinint  4536  fnresdisj  4952  fnimadisj  4962  ecinxp  6117  fzval2  8607
  Copyright terms: Public domain W3C validator