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Theorem eleqtrrid 2229
 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 2145 . 2 (𝜑𝐵 = 𝐶)
41, 3eleqtrid 2228 1 (𝜑𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331   ∈ 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:  rabsnt  3598  0elnn  4532  tfrexlem  6231  rdgtfr  6271  rdgruledefgg  6272  exmidonfinlem  7054  hashinfom  10536  ennnfonelemhom  11939  exmid1stab  13279
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