ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  3sstr3g GIF version

Theorem 3sstr3g 3048
Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
Hypotheses
Ref Expression
3sstr3g.1 (𝜑𝐴𝐵)
3sstr3g.2 𝐴 = 𝐶
3sstr3g.3 𝐵 = 𝐷
Assertion
Ref Expression
3sstr3g (𝜑𝐶𝐷)

Proof of Theorem 3sstr3g
StepHypRef Expression
1 3sstr3g.1 . 2 (𝜑𝐴𝐵)
2 3sstr3g.2 . . 3 𝐴 = 𝐶
3 3sstr3g.3 . . 3 𝐵 = 𝐷
42, 3sseq12i 3034 . 2 (𝐴𝐵𝐶𝐷)
51, 4sylib 120 1 (𝜑𝐶𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wss 2982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2988  df-ss 2995
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator