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Theorem 3sstr3g 3961
 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 3947 . 2 (𝐴𝐵𝐶𝐷)
51, 4sylib 221 1 (𝜑𝐶𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ⊆ wss 3883 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3444  df-in 3890  df-ss 3900 This theorem is referenced by:  complss  4077  uniintsn  4879  fpwwe2lem12  10071  hmeocls  22414  hmeontr  22415  usgrumgruspgr  27017  chsscon3i  29288  pjss1coi  29990  mdslmd2i  30157  satffunlem2lem2  32832  ssbnd  35377  bnd2lem  35380  trclubgNEW  40489  nzss  41192
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