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Theorem eleqtrrid 2260
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
eleqtrrid.1 𝐴𝐵
eleqtrrid.2 (𝜑𝐶 = 𝐵)
Assertion
Ref Expression
eleqtrrid (𝜑𝐴𝐶)

Proof of Theorem eleqtrrid
StepHypRef Expression
1 eleqtrrid.1 . 2 𝐴𝐵
2 eleqtrrid.2 . . 3 (𝜑𝐶 = 𝐵)
32eqcomd 2176 . 2 (𝜑𝐵 = 𝐶)
41, 3eleqtrid 2259 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wcel 2141
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-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166
This theorem is referenced by:  rabsnt  3656  0elnn  4601  canth  5805  tfrexlem  6311  rdgtfr  6351  rdgruledefgg  6352  exmidonfinlem  7163  hashinfom  10705  ennnfonelemhom  12363  exmid1stab  13998
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