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Theorem eqsstrd 3083
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
eqsstrd.1 (𝜑𝐴 = 𝐵)
eqsstrd.2 (𝜑𝐵𝐶)
Assertion
Ref Expression
eqsstrd (𝜑𝐴𝐶)

Proof of Theorem eqsstrd
StepHypRef Expression
1 eqsstrd.2 . 2 (𝜑𝐵𝐶)
2 eqsstrd.1 . . 3 (𝜑𝐴 = 𝐵)
32sseq1d 3076 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
41, 3mpbird 166 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1299  wss 3021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-in 3027  df-ss 3034
This theorem is referenced by:  eqsstr3d  3084  syl6eqss  3099  tfisi  4439  funresdfunsnss  5555  suppssof1  5930  phplem4dom  6685  cardonle  6954  exmidfodomrlemim  6966  frecuzrdgtclt  10035  ennnfonelemkh  11717  ennnfonelemf1  11723  strfvssn  11763  setscom  11781  tgrest  12120  resttopon  12122  rest0  12130  lmtopcnp  12200  metequiv2  12424  ellimc3ap  12511  dvfvalap  12523  nnsf  12783
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