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Theorem eqeltrrdi 2231
Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
eqeltrrdi.1  |-  ( ph  ->  B  =  A )
eqeltrrdi.2  |-  B  e.  C
Assertion
Ref Expression
eqeltrrdi  |-  ( ph  ->  A  e.  C )

Proof of Theorem eqeltrrdi
StepHypRef Expression
1 eqeltrrdi.1 . . 3  |-  ( ph  ->  B  =  A )
21eqcomd 2145 . 2  |-  ( ph  ->  A  =  B )
3 eqeltrrdi.2 . 2  |-  B  e.  C
42, 3eqeltrdi 2230 1  |-  ( ph  ->  A  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480
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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135
This theorem is referenced by:  eusvnfb  4375  releldm2  6083  mapprc  6546  ixpprc  6613  ixpssmap2g  6621  ixpssmapg  6622  bren  6641  brdomg  6642  mapen  6740  ssenen  6745  fi0  6863  ioof  9761  hashfacen  10586  fsum3  11163  cnrehmeocntop  12772
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