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Theorem eqsstrd 3192
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 3185 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
41, 3mpbird 167 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wss 3130
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 3136  df-ss 3143
This theorem is referenced by:  eqsstrrd  3193  eqsstrdi  3208  tfisi  4587  funresdfunsnss  5720  suppssof1  6100  phplem4dom  6862  fival  6969  fiuni  6977  cardonle  7186  exmidfodomrlemim  7200  frecuzrdgtclt  10421  ennnfonelemkh  12413  ennnfonelemf1  12419  strfvssn  12484  setscom  12502  imasaddfnlemg  12735  imasaddflemg  12737  reldvdsrsrg  13261  tgrest  13672  resttopon  13674  rest0  13682  lmtopcnp  13753  metequiv2  13999  xmettx  14013  ellimc3apf  14132  dvfvalap  14153  dvcjbr  14175  dvcj  14176  dvfre  14177  nnsf  14757
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