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Theorem eleq12i 2102
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 (A 𝐶B 𝐷)

Proof of Theorem eleq12i
StepHypRef Expression
1 eleq12i.2 . . 3 𝐶 = 𝐷
21eleq2i 2101 . 2 (A 𝐶A 𝐷)
3 eleq1i.1 . . 3 A = B
43eleq1i 2100 . 2 (A 𝐷B 𝐷)
52, 4bitri 173 1 (A 𝐶B 𝐷)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242   wcel 1390
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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033
This theorem is referenced by:  3eltr3g  2119  3eltr4g  2120  sbcel12g  2859
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