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Theorem syl6eqssr 3315
Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
syl6eqssr.1  |-  ( ph  ->  B  =  A )
syl6eqssr.2  |-  B  C_  C
Assertion
Ref Expression
syl6eqssr  |-  ( ph  ->  A  C_  C )

Proof of Theorem syl6eqssr
StepHypRef Expression
1 syl6eqssr.1 . . 3  |-  ( ph  ->  B  =  A )
21eqcomd 2371 . 2  |-  ( ph  ->  A  =  B )
3 syl6eqssr.2 . 2  |-  B  C_  C
42, 3syl6eqss 3314 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    C_ wss 3238
This theorem is referenced by:  off2  23457  cfilucfil  23802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-in 3245  df-ss 3252
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