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

Theorem pweq 3546
 Description: Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
pweq (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)

Proof of Theorem pweq
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sseq2 3152 . . 3 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21abbidv 2275 . 2 (𝐴 = 𝐵 → {𝑥𝑥𝐴} = {𝑥𝑥𝐵})
3 df-pw 3545 . 2 𝒫 𝐴 = {𝑥𝑥𝐴}
4 df-pw 3545 . 2 𝒫 𝐵 = {𝑥𝑥𝐵}
52, 3, 43eqtr4g 2215 1 (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1335  {cab 2143   ⊆ wss 3102  𝒫 cpw 3543 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115  df-pw 3545 This theorem is referenced by:  pweqi  3547  pweqd  3548  axpweq  4131  pwexg  4140  pwssunim  4243  ordpwsucexmid  4527  fival  6907  istopg  12357  istopon  12371  eltg  12412  tgdom  12432  ntrval  12470
 Copyright terms: Public domain W3C validator