Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  3eltr3i Structured version   Visualization version   GIF version

Theorem 3eltr3i 2700
 Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr3.1 𝐴𝐵
3eltr3.2 𝐴 = 𝐶
3eltr3.3 𝐵 = 𝐷
Assertion
Ref Expression
3eltr3i 𝐶𝐷

Proof of Theorem 3eltr3i
StepHypRef Expression
1 3eltr3.2 . 2 𝐴 = 𝐶
2 3eltr3.1 . . 3 𝐴𝐵
3 3eltr3.3 . . 3 𝐵 = 𝐷
42, 3eleqtri 2686 . 2 𝐴𝐷
51, 4eqeltrri 2685 1 𝐶𝐷
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606 This theorem is referenced by:  raddcn  29303  clsk1independent  37364  fourierdlem62  39061
 Copyright terms: Public domain W3C validator