Theorem pweqALT 4549

Description: Alternate proof of pweq 4548 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.

Step	Hyp	Ref	Expression
1		sseq2 3986	. . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))
2	1	abbidv 2884	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑥 ∣ 𝑥 ⊆ 𝐵})
3		df-pw 4534	. 2 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
4		df-pw 4534	. 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵}
5	2, 3, 4	3eqtr4g 2880	1 ⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)

Syntax hints:   → wi 4   = wceq 1536  {cab 2798   ⊆ wss 3929  𝒫 cpw 4532

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-in 3936  df-ss 3945  df-pw 4534

This theorem is referenced by: (None)