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Theorem eqsstrd 3191
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 3184 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
41, 3mpbird 167 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wss 3129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142
This theorem is referenced by:  eqsstrrd  3192  eqsstrdi  3207  tfisi  4586  funresdfunsnss  5719  suppssof1  6099  phplem4dom  6861  fival  6968  fiuni  6976  cardonle  7185  exmidfodomrlemim  7199  frecuzrdgtclt  10420  ennnfonelemkh  12412  ennnfonelemf1  12418  strfvssn  12483  setscom  12501  imasaddfnlemg  12734  imasaddflemg  12736  reldvdsrsrg  13259  tgrest  13639  resttopon  13641  rest0  13649  lmtopcnp  13720  metequiv2  13966  xmettx  13980  ellimc3apf  14099  dvfvalap  14120  dvcjbr  14142  dvcj  14143  dvfre  14144  nnsf  14724
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