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

Theorem eqsstrid 3138
 Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
eqsstrid.1 𝐴 = 𝐵
eqsstrid.2 (𝜑𝐵𝐶)
Assertion
Ref Expression
eqsstrid (𝜑𝐴𝐶)

Proof of Theorem eqsstrid
StepHypRef Expression
1 eqsstrid.2 . 2 (𝜑𝐵𝐶)
2 eqsstrid.1 . . 3 𝐴 = 𝐵
32sseq1i 3118 . 2 (𝐴𝐶𝐵𝐶)
41, 3sylibr 133 1 (𝜑𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331   ⊆ wss 3066 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-in 3072  df-ss 3079 This theorem is referenced by:  eqsstrrid  3139  inss  3301  difsnss  3661  tpssi  3681  peano5  4507  xpsspw  4646  iotanul  5098  iotass  5100  fun  5290  fun11iun  5381  fvss  5428  fmpt  5563  fliftrel  5686  opabbrex  5808  1stcof  6054  2ndcof  6055  tfrlemibacc  6216  tfrlemibfn  6218  tfr1onlemssrecs  6229  tfr1onlembacc  6232  tfr1onlembfn  6234  tfrcllemssrecs  6242  tfrcllembacc  6245  tfrcllembfn  6247  caucvgprlemladdrl  7479  peano5nnnn  7693  peano5nni  8716  un0addcl  9003  un0mulcl  9004  strleund  12036  cnptopco  12380  cnconst2  12391  xmetresbl  12598  blsscls2  12651  bj-omtrans  13143
 Copyright terms: Public domain W3C validator