ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  3eltr3g GIF version

Theorem 3eltr3g 2122
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
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.1 . 2 (𝜑𝐴𝐵)
2 3eltr3g.2 . . 3 𝐴 = 𝐶
3 3eltr3g.3 . . 3 𝐵 = 𝐷
42, 3eleq12i 2105 . 2 (𝐴𝐵𝐶𝐷)
51, 4sylib 127 1 (𝜑𝐶𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator