ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl5eqss GIF version

Theorem syl5eqss 3017
Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
syl5eqss.1 𝐴 = 𝐵
syl5eqss.2 (𝜑𝐵𝐶)
Assertion
Ref Expression
syl5eqss (𝜑𝐴𝐶)

Proof of Theorem syl5eqss
StepHypRef Expression
1 syl5eqss.2 . 2 (𝜑𝐵𝐶)
2 syl5eqss.1 . . 3 𝐴 = 𝐵
32sseq1i 2997 . 2 (𝐴𝐶𝐵𝐶)
41, 3sylibr 141 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2952  df-ss 2959
This theorem is referenced by:  syl5eqssr  3018  inss  3194  difsnss  3538  tpssi  3558  peano5  4349  xpsspw  4478  iotanul  4910  iotass  4912  fun  5091  fun11iun  5175  fvss  5217  fmpt  5347  fliftrel  5460  opabbrex  5577  1stcof  5818  2ndcof  5819  tfrlemibacc  5971  tfrlemibfn  5973  caucvgprlemladdrl  6834  peano5nnnn  7024  peano5nni  7993  un0addcl  8272  un0mulcl  8273  peano5setOLD  10452  bj-omtrans  10468
  Copyright terms: Public domain W3C validator