Theorem pweq 3691

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 3272	. . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))
2	1	abbidv 2358	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑥 ∣ 𝑥 ⊆ 𝐵})
3		df-pw 3690	. 2 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
4		df-pw 3690	. 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵}
5	2, 3, 4	3eqtr4g 2296	1 ⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1402  {cab 2224   ⊆ wss 3220  𝒫 cpw 3688

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233  df-pw 3690

This theorem is used by:  pweqi  3692  pweqd  3693  axpweq  4308  pwexg  4317  pwssunim  4429  ordpwsucexmid  4717  exmidpw2en  7219  fival  7304  isacnm  7559  indv  9297  hashfibc  11297  istopg  15149  istopon  15163  eltg  15202  tgdom  15222  ntrval  15260  uhgreq12g  16415  uhgr0vb  16423  isupgren  16434  isumgren  16444  isuspgren  16496  isusgren  16497  isausgren  16506