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Theorem sylan9ss 3285
 Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Hypotheses
Ref Expression
sylan9ss.1 (φA B)
sylan9ss.2 (ψB C)
Assertion
Ref Expression
sylan9ss ((φ ψ) → A C)

Proof of Theorem sylan9ss
StepHypRef Expression
1 sylan9ss.1 . 2 (φA B)
2 sylan9ss.2 . 2 (ψB C)
3 sstr 3280 . 2 ((A B B C) → A C)
41, 2, 3syl2an 463 1 ((φ ψ) → A C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ⊆ 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:  sylan9ssr  3286  psstr  3373  unss12  3435  ss2in  3482  funssxp  5233
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