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Theorem pweqALT 4612
Description: Alternate proof of pweq 4611 directly from the definition. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
pweqALT (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)

Proof of Theorem pweqALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sseq2 4003 . . 3 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21abbidv 2795 . 2 (𝐴 = 𝐵 → {𝑥𝑥𝐴} = {𝑥𝑥𝐵})
3 df-pw 4599 . 2 𝒫 𝐴 = {𝑥𝑥𝐴}
4 df-pw 4599 . 2 𝒫 𝐵 = {𝑥𝑥𝐵}
52, 3, 43eqtr4g 2791 1 (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  {cab 2703  wss 3943  𝒫 cpw 4597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-ss 3960  df-pw 4599
This theorem is referenced by: (None)
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