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Theorem eqsstrd 3305
 Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
eqsstrd.1 (φA = B)
eqsstrd.2 (φB C)
Assertion
Ref Expression
eqsstrd (φA C)

Proof of Theorem eqsstrd
StepHypRef Expression
1 eqsstrd.2 . 2 (φB C)
2 eqsstrd.1 . . 3 (φA = B)
32sseq1d 3298 . 2 (φ → (A CB C))
41, 3mpbird 223 1 (φA C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ⊆ wss 3257 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by:  eqsstr3d  3306  syl6eqss  3321  fimacnv  5411  fvmptss  5705  fvmptss2  5725  spacssnc  6284  spacind  6287
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