Theorem pweqALT 4582

Description: Alternate proof of pweq 4581 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 3966	. . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))
2	1	abbidv 2832	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑥 ∣ 𝑥 ⊆ 𝐵})
3		df-pw 4569	. 2 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
4		df-pw 4569	. 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵}
5	2, 3, 4	3eqtr4g 2826	1 ⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  {cab 2744   ⊆ wss 3908  𝒫 cpw 4567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-ss 3925  df-pw 4569

This theorem is used by: (None)