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Theorem eleq12i 2208
 Description: Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
Hypotheses
Ref Expression
eleq1i.1 𝐴 = 𝐵
eleq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
eleq12i (𝐴𝐶𝐵𝐷)

Proof of Theorem eleq12i
StepHypRef Expression
1 eleq12i.2 . . 3 𝐶 = 𝐷
21eleq2i 2207 . 2 (𝐴𝐶𝐴𝐷)
3 eleq1i.1 . . 3 𝐴 = 𝐵
43eleq1i 2206 . 2 (𝐴𝐷𝐵𝐷)
52, 4bitri 183 1 (𝐴𝐶𝐵𝐷)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1332   ∈ wcel 1481 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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-clel 2136 This theorem is referenced by:  3eltr3g  2225  3eltr4g  2226  sbcel12g  3018  ennnfonelem1  11947
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