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Theorem 3eltr4i 2199
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 2193 . 2 𝐴𝐷
51, 4eqeltri 2190 1 𝐶𝐷
Colors of variables: wff set class
Syntax hints:   = wceq 1316  wcel 1465
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-clel 2113
This theorem is referenced by:  1nq  7142  0r  7526  1sr  7527  m1r  7528
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