ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  3sstr4i Unicode version

Theorem 3sstr4i 3047
Description: Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
3sstr4.1  |-  A  C_  B
3sstr4.2  |-  C  =  A
3sstr4.3  |-  D  =  B
Assertion
Ref Expression
3sstr4i  |-  C  C_  D

Proof of Theorem 3sstr4i
StepHypRef Expression
1 3sstr4.1 . 2  |-  A  C_  B
2 3sstr4.2 . . 3  |-  C  =  A
3 3sstr4.3 . . 3  |-  D  =  B
42, 3sseq12i 3034 . 2  |-  ( C 
C_  D  <->  A  C_  B
)
51, 4mpbir 144 1  |-  C  C_  D
Colors of variables: wff set class
Syntax hints:    = wceq 1285    C_ wss 2982
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 2988  df-ss 2995
This theorem is referenced by:  undif2ss  3335  pwsnss  3615  iinuniss  3778  brab2a  4439  rncoss  4650  imassrn  4729  rnin  4783  inimass  4790  imadiflem  5029  imainlem  5031  ssoprab2i  5644  npsspw  6775  axresscn  7142
  Copyright terms: Public domain W3C validator