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Theorem pweq 3362
 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 2967 . . 3 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21abbidv 2155 . 2 (𝐴 = 𝐵 → {𝑥𝑥𝐴} = {𝑥𝑥𝐵})
3 df-pw 3361 . 2 𝒫 𝐴 = {𝑥𝑥𝐴}
4 df-pw 3361 . 2 𝒫 𝐵 = {𝑥𝑥𝐵}
52, 3, 43eqtr4g 2097 1 (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  {cab 2026   ⊆ wss 2917  𝒫 cpw 3359 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-pw 3361 This theorem is referenced by:  pweqi  3363  pweqd  3364  axpweq  3924  pwex  3932  pwexg  3933  pwssunim  4021  ordpwsucexmid  4294
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