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Theorem 3eltr3g 2855
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 2844 . 2 (𝜑𝐶𝐵)
4 3eltr3g.3 . 2 𝐵 = 𝐷
53, 4syl6eleq 2849 1 (𝜑𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-cleq 2753  df-clel 2756
This theorem is referenced by:  rankelpr  8909  isf34lem7  9393  rmulccn  30283  xrge0mulc1cn  30296  esumpfinvallem  30445  fourierdlem62  40888
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