Theorem ss2abdv

Description: Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011)

	Ref	Expression
Hypothesis	ss2abdv.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
Assertion	ss2abdv	⊢ ( 𝜑 → { 𝑥 ∣ 𝜓 } ⊆ { 𝑥 ∣ 𝜒 } )

Proof

Step	Hyp	Ref	Expression
1		ss2abdv.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2	1	alrimiv	⊢ ( 𝜑 → ∀ 𝑥 ( 𝜓 → 𝜒 ) )
3		ss2ab	⊢ ( { 𝑥 ∣ 𝜓 } ⊆ { 𝑥 ∣ 𝜒 } ↔ ∀ 𝑥 ( 𝜓 → 𝜒 ) )
4	2 3	sylibr	⊢ ( 𝜑 → { 𝑥 ∣ 𝜓 } ⊆ { 𝑥 ∣ 𝜒 } )