Theorem ss2abdv

Description: Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011) Reduce dependencies on axioms. (Revised by Steven Nguyen, 28-Jun-2024)

	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	sbimdv	⊢ ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓 → [ 𝑦 / 𝑥 ] 𝜒 ) )
3		df-clab	⊢ ( 𝑦 ∈ { 𝑥 ∣ 𝜓 } ↔ [ 𝑦 / 𝑥 ] 𝜓 )
4		df-clab	⊢ ( 𝑦 ∈ { 𝑥 ∣ 𝜒 } ↔ [ 𝑦 / 𝑥 ] 𝜒 )
5	2 3 4	3imtr4g	⊢ ( 𝜑 → ( 𝑦 ∈ { 𝑥 ∣ 𝜓 } → 𝑦 ∈ { 𝑥 ∣ 𝜒 } ) )
6	5	ssrdv	⊢ ( 𝜑 → { 𝑥 ∣ 𝜓 } ⊆ { 𝑥 ∣ 𝜒 } )