Theorem rabss2

Description: Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004) (Proof shortened by Andrew Salmon, 26-Jun-2011) Avoid axioms. (Revised by TM, 1-Feb-2026)

	Ref	Expression
Assertion	rabss2	⊢ ( 𝐴 ⊆ 𝐵 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ⊆ { 𝑥 ∈ 𝐵 ∣ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		ssel	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
2	1	anim1d	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) )
3	2	ss2abdv	⊢ ( 𝐴 ⊆ 𝐵 → { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ⊆ { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) } )
4		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
5		df-rab	⊢ { 𝑥 ∈ 𝐵 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) }
6	3 4 5	3sstr4g	⊢ ( 𝐴 ⊆ 𝐵 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ⊆ { 𝑥 ∈ 𝐵 ∣ 𝜑 } )