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Theorem sseq12d 3078
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 3076 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
3 sseq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43sseq2d 3077 . 2 (𝜑 → (𝐵𝐶𝐵𝐷))
52, 4bitrd 187 1 (𝜑 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1299  wss 3021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-in 3027  df-ss 3034
This theorem is referenced by:  3sstr3d  3091  3sstr4d  3092  ssdifeq0  3392  relcnvtr  4994  rdgisucinc  6212  oawordriexmid  6296  nnaword  6337  nnawordi  6341  sbthlem2  6774  isbth  6783  infnninf  6934  nnnninf  6935  ennnfonelemkh  11717  ennnfonelemrnh  11721  isstruct2im  11751  isstruct2r  11752  ressid2  11800  ressval2  11801  basis1  11996  baspartn  11999  eltg  12003  metss  12422  0nninf  12781  nninff  12782  nnsf  12783  peano4nninf  12784  nninfalllemn  12786  nninfalllem1  12787  nninfself  12793
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