NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  3eltr3d GIF version

Theorem 3eltr3d 2433
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr3d.1 (φA B)
3eltr3d.2 (φA = C)
3eltr3d.3 (φB = D)
Assertion
Ref Expression
3eltr3d (φC D)

Proof of Theorem 3eltr3d
StepHypRef Expression
1 3eltr3d.2 . 2 (φA = C)
2 3eltr3d.1 . . 3 (φA B)
3 3eltr3d.3 . . 3 (φB = D)
42, 3eleqtrd 2429 . 2 (φA D)
51, 4eqeltrrd 2428 1 (φC D)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1642   wcel 1710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator