Theorem abssdv

Description: Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006)

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

Proof

Step	Hyp	Ref	Expression
1		abssdv.1	⊢ ( 𝜑 → ( 𝜓 → 𝑥 ∈ 𝐴 ) )
2	1	alrimiv	⊢ ( 𝜑 → ∀ 𝑥 ( 𝜓 → 𝑥 ∈ 𝐴 ) )
3		abss	⊢ ( { 𝑥 ∣ 𝜓 } ⊆ 𝐴 ↔ ∀ 𝑥 ( 𝜓 → 𝑥 ∈ 𝐴 ) )
4	2 3	sylibr	⊢ ( 𝜑 → { 𝑥 ∣ 𝜓 } ⊆ 𝐴 )