Theorem rabssrabd

Description: Subclass of a restricted class abstraction. (Contributed by AV, 4-Jun-2022)

	Ref	Expression
Hypotheses	rabssrabd.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	rabssrabd.2	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴 ) → 𝜒 )
Assertion	rabssrabd	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } ⊆ { 𝑥 ∈ 𝐵 ∣ 𝜒 } )

Proof

Step	Hyp	Ref	Expression
1		rabssrabd.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		rabssrabd.2	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴 ) → 𝜒 )
3		3anan32	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝑥 ∈ 𝐴 ) ↔ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) )
4	3 2	sylbir	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
5	4	ex	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) )
6	5	ss2rabdv	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } ⊆ { 𝑥 ∈ 𝐴 ∣ 𝜒 } )
7		rabss2	⊢ ( 𝐴 ⊆ 𝐵 → { 𝑥 ∈ 𝐴 ∣ 𝜒 } ⊆ { 𝑥 ∈ 𝐵 ∣ 𝜒 } )
8	1 7	syl	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜒 } ⊆ { 𝑥 ∈ 𝐵 ∣ 𝜒 } )
9	6 8	sstrd	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } ⊆ { 𝑥 ∈ 𝐵 ∣ 𝜒 } )