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Theorem 3sstr3g 3608
 Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
Hypotheses
Ref Expression
3sstr3g.1 (𝜑𝐴𝐵)
3sstr3g.2 𝐴 = 𝐶
3sstr3g.3 𝐵 = 𝐷
Assertion
Ref Expression
3sstr3g (𝜑𝐶𝐷)

Proof of Theorem 3sstr3g
StepHypRef Expression
1 3sstr3g.1 . 2 (𝜑𝐴𝐵)
2 3sstr3g.2 . . 3 𝐴 = 𝐶
3 3sstr3g.3 . . 3 𝐵 = 𝐷
42, 3sseq12i 3594 . 2 (𝐴𝐵𝐶𝐷)
51, 4sylib 207 1 (𝜑𝐶𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554 This theorem is referenced by:  complss  3713  uniintsn  4449  fpwwe2lem13  9343  hmeocls  21381  hmeontr  21382  chsscon3i  27704  pjss1coi  28406  mdslmd2i  28573  ssbnd  32757  bnd2lem  32760  trclubgNEW  36944  nzss  37538
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