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Theorem pweq 3428
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 3046 . . 3 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21abbidv 2205 . 2 (𝐴 = 𝐵 → {𝑥𝑥𝐴} = {𝑥𝑥𝐵})
3 df-pw 3427 . 2 𝒫 𝐴 = {𝑥𝑥𝐴}
4 df-pw 3427 . 2 𝒫 𝐵 = {𝑥𝑥𝐵}
52, 3, 43eqtr4g 2145 1 (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  {cab 2074  wss 2997  𝒫 cpw 3425
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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010  df-pw 3427
This theorem is referenced by:  pweqi  3429  pweqd  3430  axpweq  3998  pwexg  4007  pwssunim  4102  ordpwsucexmid  4376
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