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Theorem 3sstr3g 4039
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 4025 . 2 (𝐴𝐵𝐶𝐷)
51, 4sylib 218 1 (𝜑𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wss 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-cleq 2726  df-ss 3979
This theorem is referenced by:  complss  4160  uniintsn  4989  fpwwe2lem12  10679  hmeocls  23791  hmeontr  23792  usgrumgruspgr  29213  chsscon3i  31489  pjss1coi  32191  mdslmd2i  32358  satffunlem2lem2  35390  ssbnd  37774  bnd2lem  37777  trclubgNEW  43607  nzss  44312
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