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

Theorem 3sstr4i 3211
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 3198 . 2 (𝐶𝐷𝐴𝐵)
51, 4mpbir 146 1 𝐶𝐷
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wss 3144
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157
This theorem is referenced by:  undif2ss  3513  pwsnss  3818  iinuniss  3984  brab2a  4697  rncoss  4915  imassrn  4999  rnin  5056  inimass  5063  imadiflem  5314  imainlem  5316  ssoprab2i  5984  npsspw  7499  axresscn  7888
  Copyright terms: Public domain W3C validator