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Theorem pweq 3479
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 3087 . . 3 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21abbidv 2232 . 2 (𝐴 = 𝐵 → {𝑥𝑥𝐴} = {𝑥𝑥𝐵})
3 df-pw 3478 . 2 𝒫 𝐴 = {𝑥𝑥𝐴}
4 df-pw 3478 . 2 𝒫 𝐵 = {𝑥𝑥𝐵}
52, 3, 43eqtr4g 2172 1 (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  {cab 2101  wss 3037  𝒫 cpw 3476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-in 3043  df-ss 3050  df-pw 3478
This theorem is referenced by:  pweqi  3480  pweqd  3481  axpweq  4055  pwexg  4064  pwssunim  4166  ordpwsucexmid  4445  fival  6810  istopg  12009  istopon  12023  eltg  12064  tgdom  12084  ntrval  12122
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