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Theorem sseq12d 3128
Description: An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
Hypotheses
Ref Expression
sseq1d.1 (𝜑𝐴 = 𝐵)
sseq12d.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
sseq12d (𝜑 → (𝐴𝐶𝐵𝐷))

Proof of Theorem sseq12d
StepHypRef Expression
1 sseq1d.1 . . 3 (𝜑𝐴 = 𝐵)
21sseq1d 3126 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
3 sseq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43sseq2d 3127 . 2 (𝜑 → (𝐵𝐶𝐵𝐷))
52, 4bitrd 187 1 (𝜑 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = 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:  3sstr3d  3141  3sstr4d  3142  ssdifeq0  3445  relcnvtr  5058  rdgisucinc  6282  oawordriexmid  6366  nnaword  6407  nnawordi  6411  sbthlem2  6846  isbth  6855  infnninf  7022  nnnninf  7023  ennnfonelemkh  11936  ennnfonelemrnh  11940  isstruct2im  11983  isstruct2r  11984  ressid2  12032  ressval2  12033  basis1  12228  baspartn  12231  eltg  12235  metss  12677  0nninf  13258  nninff  13259  nnsf  13260  peano4nninf  13261  nninfalllemn  13263  nninfalllem1  13264  nninfself  13270
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