Theorem pweq 3609

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 3208	. . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))
2	1	abbidv 2314	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑥 ∣ 𝑥 ⊆ 𝐵})
3		df-pw 3608	. 2 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
4		df-pw 3608	. 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵}
5	2, 3, 4	3eqtr4g 2254	1 ⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)

Syntax hints:   → wi 4   = wceq 1364  {cab 2182   ⊆ wss 3157  𝒫 cpw 3606

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170  df-pw 3608

This theorem is referenced by:  pweqi  3610  pweqd  3611  axpweq  4205  pwexg  4214  pwssunim  4320  ordpwsucexmid  4607  exmidpw2en  6982  fival  7045  isacnm  7286  istopg  14319  istopon  14333  eltg  14372  tgdom  14392  ntrval  14430