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Theorem syl5eqss 3315
 Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
syl5eqss.1 A = B
syl5eqss.2 (φB C)
Assertion
Ref Expression
syl5eqss (φA C)

Proof of Theorem syl5eqss
StepHypRef Expression
1 syl5eqss.2 . 2 (φB C)
2 syl5eqss.1 . . 3 A = B
32sseq1i 3295 . 2 (A CB C)
41, 3sylibr 203 1 (φA C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ⊆ wss 3257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  syl5eqssr  3316  inss  3484  peano5  4409  spfininduct  4540  fun  5236  fmpt  5692  clos1induct  5880  sbthlem1  6203  spacssnc  6284  frecxp  6314  frecxpg  6315
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