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Theorem sylan9ss 2952
 Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Hypotheses
Ref Expression
sylan9ss.1 (φAB)
sylan9ss.2 (ψB𝐶)
Assertion
Ref Expression
sylan9ss ((φ ψ) → A𝐶)

Proof of Theorem sylan9ss
StepHypRef Expression
1 sylan9ss.1 . 2 (φAB)
2 sylan9ss.2 . 2 (ψB𝐶)
3 sstr 2947 . 2 ((AB B𝐶) → A𝐶)
41, 2, 3syl2an 273 1 ((φ ψ) → A𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ⊆ wss 2911 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925 This theorem is referenced by:  sylan9ssr  2953  psstr  3043  sspsstr  3044  psssstr  3045  unss12  3109  ss2in  3158  relrelss  4787  funssxp  5003
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