Theorem rabssdv

Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 2-Feb-2015)

	Ref	Expression
Hypothesis	rabssdv.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓 ) → 𝑥 ∈ 𝐵 )
Assertion	rabssdv	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		rabssdv.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓 ) → 𝑥 ∈ 𝐵 )
2	1	3exp	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝑥 ∈ 𝐵 ) ) )
3	2	ralrimiv	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝑥 ∈ 𝐵 ) )
4		rabss	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜓 } ⊆ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝑥 ∈ 𝐵 ) )
5	3 4	sylibr	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } ⊆ 𝐵 )