Theorem pweq 3725

Description: Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
pweq	⊢ (A = B → ℘A = ℘B)

Proof of Theorem pweq

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 3293	. . 3 ⊢ (A = B → (x ⊆ A ↔ x ⊆ B))
2	1	abbidv 2467	. 2 ⊢ (A = B → {x ∣ x ⊆ A} = {x ∣ x ⊆ B})
3		df-pw 3724	. 2 ⊢ ℘A = {x ∣ x ⊆ A}
4		df-pw 3724	. 2 ⊢ ℘B = {x ∣ x ⊆ B}
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → ℘A = ℘B)

Syntax hints:   → wi 4   = wceq 1642  {cab 2339   ⊆ wss 3257  ℘cpw 3722

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pw 3724

This theorem is referenced by:  pweqi  3726  pweqd  3727  pw1eq  4143  nnpweq  4523  sfin01  4528  sfindbl  4530  1cvsfin  4542  vfinspsslem1  4550  enpw  6087