Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  3eltr4g GIF version

Theorem 3eltr4g 2227
 Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
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.1 . 2 (𝜑𝐴𝐵)
2 3eltr4g.2 . . 3 𝐶 = 𝐴
3 3eltr4g.3 . . 3 𝐷 = 𝐵
42, 3eleq12i 2209 . 2 (𝐶𝐷𝐴𝐵)
51, 4sylibr 133 1 (𝜑𝐶𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 2112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1487  ax-17 1503  ax-ial 1511  ax-ext 2123 This theorem depends on definitions:  df-bi 116  df-cleq 2134  df-clel 2137 This theorem is referenced by:  riotacl2  5755  2strop1g  12139
 Copyright terms: Public domain W3C validator