Theorem pweq

Description: Equality theorem for power class. (Contributed by NM, 21-Jun-1993) (Proof shortened by BJ, 13-Apr-2024)

	Ref	Expression
Assertion	pweq	⊢ ( 𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		eqimss	⊢ ( 𝐴 = 𝐵 → 𝐴 ⊆ 𝐵 )
2	1	sspwd	⊢ ( 𝐴 = 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵 )
3		eqimss2	⊢ ( 𝐴 = 𝐵 → 𝐵 ⊆ 𝐴 )
4	3	sspwd	⊢ ( 𝐴 = 𝐵 → 𝒫 𝐵 ⊆ 𝒫 𝐴 )
5	2 4	eqssd	⊢ ( 𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵 )