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Theorem sseq12d 3039
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 3037 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
3 sseq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43sseq2d 3038 . 2 (𝜑 → (𝐵𝐶𝐵𝐷))
52, 4bitrd 186 1 (𝜑 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wss 2984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2990  df-ss 2997
This theorem is referenced by:  3sstr3d  3052  3sstr4d  3053  ssdifeq0  3346  relcnvtr  4904  rdgisucinc  6082  oawordriexmid  6163  nnaword  6200  nnawordi  6204  infnninf  8644  nnnninf  8645
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