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Theorem pweq 3337
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 2944 . . 3 (A = B → (xAxB))
21abbidv 2137 . 2 (A = B → {xxA} = {xxB})
3 df-pw 3336 . 2 𝒫 A = {xxA}
4 df-pw 3336 . 2 𝒫 B = {xxB}
52, 3, 43eqtr4g 2079 1 (A = B → 𝒫 A = 𝒫 B)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228  {cab 2008  wss 2894  𝒫 cpw 3334
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-in 2901  df-ss 2908  df-pw 3336
This theorem is referenced by:  pweqi  3338  pweqd  3339  axpweq  3898  pwex  3906  pwexg  3907  pwssunim  3995  ordpwsucexmid  4230
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