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Theorem eqsstrdi 3222
Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
eqsstrdi.1  |-  ( ph  ->  A  =  B )
eqsstrdi.2  |-  B  C_  C
Assertion
Ref Expression
eqsstrdi  |-  ( ph  ->  A  C_  C )

Proof of Theorem eqsstrdi
StepHypRef Expression
1 eqsstrdi.1 . 2  |-  ( ph  ->  A  =  B )
2 eqsstrdi.2 . . 3  |-  B  C_  C
32a1i 9 . 2  |-  ( ph  ->  B  C_  C )
41, 3eqsstrd 3206 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    C_ wss 3144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157
This theorem is referenced by:  eqsstrrdi  3223  resasplitss  5410  fimacnv  5661  en2other2  7213  exmidfodomrlemim  7218  pw1on  7243  suplocexprlemex  7739  1arith  12383  ennnfonelemkh  12431  aprap  13563  toponsspwpwg  13919  ntrss2  14018  cnprcl2k  14103  reldvg  14545  bj-nntrans  15100  nninfsellemsuc  15159
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