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Theorem 3eltr4i 2135
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 2129 . 2 𝐴𝐷
51, 4eqeltri 2126 1 𝐶𝐷
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-clel 2052
This theorem is referenced by:  1nq  6522  0r  6893  1sr  6894  m1r  6895
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