Theorem ss2rabdf

Description: Deduction of restricted abstraction subclass from implication. (Contributed by Glauco Siliprandi, 21-Dec-2024)

	Ref	Expression
Hypotheses	ss2rabdf.1	⊢ Ⅎ 𝑥 𝜑
	ss2rabdf.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) )
Assertion	ss2rabdf	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } ⊆ { 𝑥 ∈ 𝐴 ∣ 𝜒 } )

Proof

Step	Hyp	Ref	Expression
1		ss2rabdf.1	⊢ Ⅎ 𝑥 𝜑
2		ss2rabdf.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) )
3	2	ex	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) )
4	1 3	ralrimi	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )
5		ss2rab	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜓 } ⊆ { 𝑥 ∈ 𝐴 ∣ 𝜒 } ↔ ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )
6	4 5	sylibr	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } ⊆ { 𝑥 ∈ 𝐴 ∣ 𝜒 } )