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Theorem 3eltr4i 2119
 Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr4.1 𝐴𝐵
3eltr4.2 𝐶 = 𝐴
3eltr4.3 𝐷 = 𝐵
Assertion
Ref Expression
3eltr4i 𝐶𝐷

Proof of Theorem 3eltr4i
StepHypRef Expression
1 3eltr4.2 . 2 𝐶 = 𝐴
2 3eltr4.1 . . 3 𝐴𝐵
3 3eltr4.3 . . 3 𝐷 = 𝐵
42, 3eleqtrri 2113 . 2 𝐴𝐷
51, 4eqeltri 2110 1 𝐶𝐷
 Colors of variables: wff set class Syntax hints:   = 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:  1nq  6464  0r  6835  1sr  6836  m1r  6837
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