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Theorem sseqtrd 3063
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
sseqtrd.1 (𝜑𝐴𝐵)
sseqtrd.2 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
sseqtrd (𝜑𝐴𝐶)

Proof of Theorem sseqtrd
StepHypRef Expression
1 sseqtrd.1 . 2 (𝜑𝐴𝐵)
2 sseqtrd.2 . . 3 (𝜑𝐵 = 𝐶)
32sseq2d 3055 . 2 (𝜑 → (𝐴𝐵𝐴𝐶))
41, 3mpbid 146 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1290  wss 3000
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 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-in 3006  df-ss 3013
This theorem is referenced by:  sseqtr4d  3064  fssdmd  5187  resasplitss  5203  nnaword2  6287  erssxp  6329  phpm  6635  ioodisj  9464  tgcl  11818  basgen  11834  bastop1  11837  bastop2  11838  clsss2  11883  nninfalllemn  12164
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