Theorem pweq 3655

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.

Step	Hyp	Ref	Expression
1		sseq2 3251	. . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))
2	1	abbidv 2349	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑥 ∣ 𝑥 ⊆ 𝐵})
3		df-pw 3654	. 2 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
4		df-pw 3654	. 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵}
5	2, 3, 4	3eqtr4g 2289	1 ⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)

Syntax hints:   → wi 4   = wceq 1397  {cab 2217   ⊆ wss 3200  𝒫 cpw 3652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213  df-pw 3654

This theorem is referenced by:  pweqi  3656  pweqd  3657  axpweq  4261  pwexg  4270  pwssunim  4381  ordpwsucexmid  4668  exmidpw2en  7104  fival  7169  isacnm  7418  istopg  14742  istopon  14756  eltg  14795  tgdom  14815  ntrval  14853  uhgreq12g  15946  uhgr0vb  15954  isupgren  15965  isumgren  15975  isuspgren  16027  isusgren  16028  isausgren  16037