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Theorem 3sstr3g 3864
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 3850 . 2 (𝐴𝐵𝐶𝐷)
51, 4sylib 210 1 (𝜑𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wss 3792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-in 3799  df-ss 3806
This theorem is referenced by:  complss  3974  uniintsn  4747  fpwwe2lem13  9799  hmeocls  21980  hmeontr  21981  usgrumgruspgr  26529  chsscon3i  28892  pjss1coi  29594  mdslmd2i  29761  ssbnd  34213  bnd2lem  34216  trclubgNEW  38886  nzss  39476
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