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Theorem sseqtrrd 3196
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 2183 . 2 (𝜑𝐵 = 𝐶)
41, 3sseqtrd 3195 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wss 3131
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144
This theorem is referenced by:  sseqtrrid  3208  fnfvima  5753  tfrlemiubacc  6333  tfr1onlemubacc  6349  tfrcllemubacc  6362  rdgivallem  6384  nnnninf  7126  nninfwlpoimlemg  7175  dfphi2  12222  ctinf  12433  imasaddfnlemg  12740  imasaddvallemg  12741  subsubg  13062  subsubrg  13371  toponss  13611  ssntr  13707  iscnp3  13788  cnprcl2k  13791  tgcn  13793  tgcnp  13794  ssidcn  13795  cncnp  13815  txcnp  13856  imasnopn  13884  hmeontr  13898  blssec  14023  blssopn  14070  xmettx  14095  metcnp  14097  nnsf  14839  nninfsellemsuc  14846
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