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Theorem 3eltr4g 2715
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
3eltr4g.1 (𝜑𝐴𝐵)
3eltr4g.2 𝐶 = 𝐴
3eltr4g.3 𝐷 = 𝐵
Assertion
Ref Expression
3eltr4g (𝜑𝐶𝐷)

Proof of Theorem 3eltr4g
StepHypRef Expression
1 3eltr4g.2 . . 3 𝐶 = 𝐴
2 3eltr4g.1 . . 3 (𝜑𝐴𝐵)
31, 2syl5eqel 2702 . 2 (𝜑𝐶𝐵)
4 3eltr4g.3 . 2 𝐷 = 𝐵
53, 4syl6eleqr 2709 1 (𝜑𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-cleq 2614  df-clel 2617
This theorem is referenced by:  riotacl2  6589  rankelun  8695  rankelpr  8696  rankelop  8697  cdivcncf  22660  itg1addlem4  23406  cxpcn3  24423  bposlem4  24946  mirauto  25513  ldgenpisyslem1  30049  nosepdm  31621  relowlpssretop  32883  mapfzcons  36798  fourierdlem62  39722  fourierdlem63  39723
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