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

Theorem eqsstrrdi 3150
Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
eqsstrrdi.1  |-  ( ph  ->  B  =  A )
eqsstrrdi.2  |-  B  C_  C
Assertion
Ref Expression
eqsstrrdi  |-  ( ph  ->  A  C_  C )

Proof of Theorem eqsstrrdi
StepHypRef Expression
1 eqsstrrdi.1 . . 3  |-  ( ph  ->  B  =  A )
21eqcomd 2145 . 2  |-  ( ph  ->  A  =  B )
3 eqsstrrdi.2 . 2  |-  B  C_  C
42, 3eqsstrdi 3149 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    C_ 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:  ffvresb  5583  tposss  6143  sbthlemi5  6849  iooval2  9705  telfsumo  11242  structcnvcnv  11984  txss12  12444  txbasval  12445
  Copyright terms: Public domain W3C validator