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

Theorem sseqtrrd 3136
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
sseqtrrd.1 (𝜑𝐴𝐵)
sseqtrrd.2 (𝜑𝐶 = 𝐵)
Assertion
Ref Expression
sseqtrrd (𝜑𝐴𝐶)

Proof of Theorem sseqtrrd
StepHypRef Expression
1 sseqtrrd.1 . 2 (𝜑𝐴𝐵)
2 sseqtrrd.2 . . 3 (𝜑𝐶 = 𝐵)
32eqcomd 2145 . 2 (𝜑𝐵 = 𝐶)
41, 3sseqtrd 3135 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wss 3071
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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084
This theorem is referenced by:  sseqtrrid  3148  fnfvima  5652  tfrlemiubacc  6227  tfr1onlemubacc  6243  tfrcllemubacc  6256  rdgivallem  6278  nnnninf  7023  dfphi2  11903  ctinf  11950  toponss  12203  ssntr  12301  iscnp3  12382  cnprcl2k  12385  tgcn  12387  tgcnp  12388  ssidcn  12389  cncnp  12409  txcnp  12450  imasnopn  12478  hmeontr  12492  blssec  12617  blssopn  12664  xmettx  12689  metcnp  12691  nnsf  13213  nninfsellemsuc  13222
  Copyright terms: Public domain W3C validator