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Theorem eleq12i 2105
 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 2104 . 2 (𝐴𝐶𝐴𝐷)
3 eleq1i.1 . . 3 𝐴 = 𝐵
43eleq1i 2103 . 2 (𝐴𝐷𝐵𝐷)
52, 4bitri 173 1 (𝐴𝐶𝐵𝐷)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1243   ∈ wcel 1393 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036 This theorem is referenced by:  3eltr3g  2122  3eltr4g  2123  sbcel12g  2865
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