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

Theorem pweq 3390
 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 2995 . . 3 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21abbidv 2171 . 2 (𝐴 = 𝐵 → {𝑥𝑥𝐴} = {𝑥𝑥𝐵})
3 df-pw 3389 . 2 𝒫 𝐴 = {𝑥𝑥𝐴}
4 df-pw 3389 . 2 𝒫 𝐵 = {𝑥𝑥𝐵}
52, 3, 43eqtr4g 2113 1 (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259  {cab 2042   ⊆ wss 2945  𝒫 cpw 3387 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2952  df-ss 2959  df-pw 3389 This theorem is referenced by:  pweqi  3391  pweqd  3392  axpweq  3952  pwex  3960  pwexg  3961  pwssunim  4049  ordpwsucexmid  4322
 Copyright terms: Public domain W3C validator