Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sseq2i GIF version

Theorem sseq2i 3124
 Description: An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
sseq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
sseq2i (𝐶𝐴𝐶𝐵)

Proof of Theorem sseq2i
StepHypRef Expression
1 sseq1i.1 . 2 𝐴 = 𝐵
2 sseq2 3121 . 2 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
31, 2ax-mp 5 1 (𝐶𝐴𝐶𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1331   ⊆ wss 3071 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 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084 This theorem is referenced by:  sseqtri  3131  sseqtrdi  3145  abss  3166  ssrab  3175  ssintrab  3794  iunpwss  3904  iotass  5105  dffun2  5133  ssimaex  5482  isstructim  11987  isstructr  11988  bj-ssom  13218  ss1oel2o  13273
 Copyright terms: Public domain W3C validator