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Theorem syl6eqss 3241
Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
syl6eqss.1  |-  ( ph  ->  A  =  B )
syl6eqss.2  |-  B  C_  C
Assertion
Ref Expression
syl6eqss  |-  ( ph  ->  A  C_  C )

Proof of Theorem syl6eqss
StepHypRef Expression
1 syl6eqss.1 . 2  |-  ( ph  ->  A  =  B )
2 syl6eqss.2 . . 3  |-  B  C_  C
32a1i 10 . 2  |-  ( ph  ->  B  C_  C )
41, 3eqsstrd 3225 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    C_ wss 3165
This theorem is referenced by:  syl6eqssr  3242  sectss  13671  invss  13679  fullfunc  13796  fthfunc  13797  catccatid  13950  resscatc  13953  catcisolem  13954  catciso  13955  yoniso  14075  ssnnssfz  23292  gsumpropd2lem  23394  nbgrassvt  28282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179
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