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Theorem syl6sseqr 3071
Description: A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
syl6ssr.1  |-  ( ph  ->  A  C_  B )
syl6ssr.2  |-  C  =  B
Assertion
Ref Expression
syl6sseqr  |-  ( ph  ->  A  C_  C )

Proof of Theorem syl6sseqr
StepHypRef Expression
1 syl6ssr.1 . 2  |-  ( ph  ->  A  C_  B )
2 syl6ssr.2 . . 3  |-  C  =  B
32eqcomi 2092 . 2  |-  B  =  C
41, 3syl6sseq 3070 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    C_ wss 2997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010
This theorem is referenced by:  iunpw  4292  iotanul  4982  iotass  4984  tfrlem9  6066  tfrlemibfn  6075  tfrlemiubacc  6077  tfrlemi14d  6080  tfr1onlemssrecs  6086  tfr1onlemres  6096  tfrcllemres  6109  exmidfodomrlemr  6807  exmidfodomrlemrALT  6808  uznnssnn  9034  shftfvalg  10217  shftfval  10220
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