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Theorem 3eltr3g 2894
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by Wolf Lammen, 23-Nov-2019.)
Hypotheses
Ref Expression
3eltr3g.1 (𝜑𝐴𝐵)
3eltr3g.2 𝐴 = 𝐶
3eltr3g.3 𝐵 = 𝐷
Assertion
Ref Expression
3eltr3g (𝜑𝐶𝐷)

Proof of Theorem 3eltr3g
StepHypRef Expression
1 3eltr3g.2 . . 3 𝐴 = 𝐶
2 3eltr3g.1 . . 3 (𝜑𝐴𝐵)
31, 2syl5eqelr 2883 . 2 (𝜑𝐶𝐵)
4 3eltr3g.3 . 2 𝐵 = 𝐷
53, 4syl6eleq 2888 1 (𝜑𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-cleq 2792  df-clel 2795
This theorem is referenced by:  rankelpr  8986  isf34lem7  9489  rmulccn  30490  xrge0mulc1cn  30503  esumpfinvallem  30652  fourierdlem62  41128
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