MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pweqALT Structured version   Visualization version   GIF version

Theorem pweqALT 4615
Description: Alternate proof of pweq 4614 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 4010 . . 3 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21abbidv 2808 . 2 (𝐴 = 𝐵 → {𝑥𝑥𝐴} = {𝑥𝑥𝐵})
3 df-pw 4602 . 2 𝒫 𝐴 = {𝑥𝑥𝐴}
4 df-pw 4602 . 2 𝒫 𝐵 = {𝑥𝑥𝐵}
52, 3, 43eqtr4g 2802 1 (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  {cab 2714  wss 3951  𝒫 cpw 4600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-ss 3968  df-pw 4602
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator