Theorem ralrab

Description: Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010)

	Ref	Expression
Hypothesis	ralab.1	⊢ ( 𝑦 = 𝑥 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ralrab	⊢ ( ∀ 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } 𝜒 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralab.1	⊢ ( 𝑦 = 𝑥 → ( 𝜑 ↔ 𝜓 ) )
2	1	elrab	⊢ ( 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
3	2	imbi1i	⊢ ( ( 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } → 𝜒 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) → 𝜒 ) )
4		impexp	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) → 𝜒 ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) )
5	3 4	bitri	⊢ ( ( 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } → 𝜒 ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) )
6	5	ralbii2	⊢ ( ∀ 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } 𝜒 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )