Theorem ss2rabdv

Description: Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006)

	Ref	Expression
Hypothesis	ss2rabdv.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) )
Assertion	ss2rabdv	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } ⊆ { 𝑥 ∈ 𝐴 ∣ 𝜒 } )

Proof

Step	Hyp	Ref	Expression
1		ss2rabdv.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) )
2	1	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )
3		ss2rab	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜓 } ⊆ { 𝑥 ∈ 𝐴 ∣ 𝜒 } ↔ ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )
4	2 3	sylibr	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } ⊆ { 𝑥 ∈ 𝐴 ∣ 𝜒 } )