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Theorem sseq12 3788
Description: Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
Assertion
Ref Expression
sseq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))

Proof of Theorem sseq12
StepHypRef Expression
1 sseq1 3786 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
2 sseq2 3787 . 2 (𝐶 = 𝐷 → (𝐵𝐶𝐵𝐷))
31, 2sylan9bb 505 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wss 3732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-in 3739  df-ss 3746
This theorem is referenced by:  sseq12i  3791  sorpsscmpl  7146  funcnvuni  7317  fun11iun  7324  sornom  9352  axdc3lem2  9526  ipole  17424  ipodrsima  17431  metsscmetcld  23392  funpsstri  32108  brreds  34796  ismrcd2  37940  ismrc  37942
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