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Theorem syl6eqss 3321
 Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
syl6eqss.1 (φA = B)
syl6eqss.2 B C
Assertion
Ref Expression
syl6eqss (φA C)

Proof of Theorem syl6eqss
StepHypRef Expression
1 syl6eqss.1 . 2 (φA = B)
2 syl6eqss.2 . . 3 B C
32a1i 10 . 2 (φB C)
41, 3eqsstrd 3305 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:  syl6eqssr  3322  dmsnopss  5067  frecxp  6314
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