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

Theorem eqsstrd 3103
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
eqsstrd.1 (𝜑𝐴 = 𝐵)
eqsstrd.2 (𝜑𝐵𝐶)
Assertion
Ref Expression
eqsstrd (𝜑𝐴𝐶)

Proof of Theorem eqsstrd
StepHypRef Expression
1 eqsstrd.2 . 2 (𝜑𝐵𝐶)
2 eqsstrd.1 . . 3 (𝜑𝐴 = 𝐵)
32sseq1d 3096 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
41, 3mpbird 166 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wss 3041
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-in 3047  df-ss 3054
This theorem is referenced by:  eqsstrrd  3104  eqsstrdi  3119  tfisi  4471  funresdfunsnss  5591  suppssof1  5967  phplem4dom  6724  fival  6826  fiuni  6834  cardonle  7011  exmidfodomrlemim  7025  frecuzrdgtclt  10162  ennnfonelemkh  11852  ennnfonelemf1  11858  strfvssn  11908  setscom  11926  tgrest  12265  resttopon  12267  rest0  12275  lmtopcnp  12346  metequiv2  12592  xmettx  12606  ellimc3apf  12725  dvfvalap  12746  dvcjbr  12768  dvcj  12769  dvfre  12770  nnsf  13126
  Copyright terms: Public domain W3C validator