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

Theorem eqsstrrd 3192
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
eqsstrrd.1  |-  ( ph  ->  B  =  A )
eqsstrrd.2  |-  ( ph  ->  B  C_  C )
Assertion
Ref Expression
eqsstrrd  |-  ( ph  ->  A  C_  C )

Proof of Theorem eqsstrrd
StepHypRef Expression
1 eqsstrrd.1 . . 3  |-  ( ph  ->  B  =  A )
21eqcomd 2183 . 2  |-  ( ph  ->  A  =  B )
3 eqsstrrd.2 . 2  |-  ( ph  ->  B  C_  C )
42, 3eqsstrd 3191 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    C_ wss 3129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142
This theorem is referenced by:  ssxpbm  5059  ssxp1  5060  ssxp2  5061  suppssof1  6093  tfrlemiubacc  6324  tfr1onlemubacc  6340  tfrcllemubacc  6353  oaword1  6465  phplem4dom  6855  fisseneq  6924  nnnninfeq2  7120  archnqq  7394  ringidss  13025  epttop  13223  metequiv2  13629  limccnpcntop  13777  limccnp2lem  13778  limccnp2cntop  13779  nnsf  14377
  Copyright terms: Public domain W3C validator