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Theorem eqsstrrdi 3200
Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
eqsstrrdi.1  |-  ( ph  ->  B  =  A )
eqsstrrdi.2  |-  B  C_  C
Assertion
Ref Expression
eqsstrrdi  |-  ( ph  ->  A  C_  C )

Proof of Theorem eqsstrrdi
StepHypRef Expression
1 eqsstrrdi.1 . . 3  |-  ( ph  ->  B  =  A )
21eqcomd 2176 . 2  |-  ( ph  ->  A  =  B )
3 eqsstrrdi.2 . 2  |-  B  C_  C
42, 3eqsstrdi 3199 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    C_ wss 3121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134
This theorem is referenced by:  ffvresb  5656  tposss  6222  sbthlemi5  6934  iooval2  9859  telfsumo  11416  structcnvcnv  12419  txss12  13019  txbasval  13020
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