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

Theorem 3sstr4i 3039
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 𝐴𝐵
3sstr4.2 𝐶 = 𝐴
3sstr4.3 𝐷 = 𝐵
Assertion
Ref Expression
3sstr4i 𝐶𝐷

Proof of Theorem 3sstr4i
StepHypRef Expression
1 3sstr4.1 . 2 𝐴𝐵
2 3sstr4.2 . . 3 𝐶 = 𝐴
3 3sstr4.3 . . 3 𝐷 = 𝐵
42, 3sseq12i 3026 . 2 (𝐶𝐷𝐴𝐵)
51, 4mpbir 144 1 𝐶𝐷
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wss 2974
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 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987
This theorem is referenced by:  undif2ss  3326  pwsnss  3603  iinuniss  3766  brab2a  4419  rncoss  4630  imassrn  4709  rnin  4763  inimass  4770  imadiflem  5009  imainlem  5011  ssoprab2i  5624  npsspw  6723  axresscn  7090
  Copyright terms: Public domain W3C validator