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

Theorem sseq2i 3035
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  |-  ( C 
C_  A  <->  C  C_  B
)

Proof of Theorem sseq2i
StepHypRef Expression
1 sseq1i.1 . 2  |-  A  =  B
2 sseq2 3032 . 2  |-  ( A  =  B  ->  ( C  C_  A  <->  C  C_  B
) )
31, 2ax-mp 7 1  |-  ( C 
C_  A  <->  C  C_  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1285    C_ wss 2984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2990  df-ss 2997
This theorem is referenced by:  sseqtri  3042  syl6sseq  3056  abss  3074  ssrab  3083  ssintrab  3685  iunpwss  3790  iotass  4951  dffun2  4979  ssimaex  5310  bj-ssom  11174
  Copyright terms: Public domain W3C validator