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Theorem 3sstr3g 4036
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.2 . . 3 𝐴 = 𝐶
2 3sstr3g.1 . . 3 (𝜑𝐴𝐵)
31, 2eqsstrrid 4023 . 2 (𝜑𝐶𝐵)
4 3sstr3g.3 . 2 𝐵 = 𝐷
53, 4sseqtrdi 4024 1 (𝜑𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-ss 3968
This theorem is referenced by:  complss  4151  uniintsn  4985  fpwwe2lem12  10682  hmeocls  23776  hmeontr  23777  usgrumgruspgr  29199  chsscon3i  31480  pjss1coi  32182  mdslmd2i  32349  satffunlem2lem2  35411  ssbnd  37795  bnd2lem  37798  trclubgNEW  43631  nzss  44336
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