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Theorem syl6eqss 3228
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 3212 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    C_ wss 3152
This theorem is referenced by:  syl6eqssr  3229  sectss  13655  invss  13663  fullfunc  13780  fthfunc  13781  catccatid  13934  resscatc  13937  catcisolem  13938  catciso  13939  yoniso  14059  ssnnssfz  23277  gsumpropd2lem  23379  nbgrassvt  28148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166
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