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Theorem eqsstrd 3004
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 2997 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
41, 3mpbird 160 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1257  wss 2942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-11 1411  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-in 2949  df-ss 2956
This theorem is referenced by:  eqsstr3d  3005  syl6eqss  3020  tfisi  4335  suppssof1  5753  phplem4dom  6352  cardonle  6395
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