Theorem ssintrab

Description: Subclass of the intersection of a restricted class abstraction. (Contributed by NM, 30-Jan-2015)

	Ref	Expression
Assertion	ssintrab	⊢ ( 𝐴 ⊆ ∩ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ↔ ∀ 𝑥 ∈ 𝐵 ( 𝜑 → 𝐴 ⊆ 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		df-rab	⊢ { 𝑥 ∈ 𝐵 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) }
2	1	inteqi	⊢ ∩ { 𝑥 ∈ 𝐵 ∣ 𝜑 } = ∩ { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) }
3	2	sseq2i	⊢ ( 𝐴 ⊆ ∩ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ↔ 𝐴 ⊆ ∩ { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) } )
4		impexp	⊢ ( ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝐴 ⊆ 𝑥 ) ↔ ( 𝑥 ∈ 𝐵 → ( 𝜑 → 𝐴 ⊆ 𝑥 ) ) )
5	4	albii	⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝐴 ⊆ 𝑥 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → ( 𝜑 → 𝐴 ⊆ 𝑥 ) ) )
6		ssintab	⊢ ( 𝐴 ⊆ ∩ { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) } ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) → 𝐴 ⊆ 𝑥 ) )
7		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐵 ( 𝜑 → 𝐴 ⊆ 𝑥 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → ( 𝜑 → 𝐴 ⊆ 𝑥 ) ) )
8	5 6 7	3bitr4i	⊢ ( 𝐴 ⊆ ∩ { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) } ↔ ∀ 𝑥 ∈ 𝐵 ( 𝜑 → 𝐴 ⊆ 𝑥 ) )
9	3 8	bitri	⊢ ( 𝐴 ⊆ ∩ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ↔ ∀ 𝑥 ∈ 𝐵 ( 𝜑 → 𝐴 ⊆ 𝑥 ) )