Theorem ralrab2

Description: Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015)

	Ref	Expression
Hypothesis	ralab2.1	⊢ ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) )
Assertion	ralrab2	⊢ ( ∀ 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } 𝜓 ↔ ∀ 𝑦 ∈ 𝐴 ( 𝜑 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralab2.1	⊢ ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) )
2		df-rab	⊢ { 𝑦 ∈ 𝐴 ∣ 𝜑 } = { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) }
3	2	raleqi	⊢ ( ∀ 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } 𝜓 ↔ ∀ 𝑥 ∈ { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) } 𝜓 )
4	1	ralab2	⊢ ( ∀ 𝑥 ∈ { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) } 𝜓 ↔ ∀ 𝑦 ( ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) → 𝜒 ) )
5		impexp	⊢ ( ( ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) → 𝜒 ) ↔ ( 𝑦 ∈ 𝐴 → ( 𝜑 → 𝜒 ) ) )
6	5	albii	⊢ ( ∀ 𝑦 ( ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) → 𝜒 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → ( 𝜑 → 𝜒 ) ) )
7		df-ral	⊢ ( ∀ 𝑦 ∈ 𝐴 ( 𝜑 → 𝜒 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → ( 𝜑 → 𝜒 ) ) )
8	6 7	bitr4i	⊢ ( ∀ 𝑦 ( ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) → 𝜒 ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝜑 → 𝜒 ) )
9	3 4 8	3bitri	⊢ ( ∀ 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } 𝜓 ↔ ∀ 𝑦 ∈ 𝐴 ( 𝜑 → 𝜒 ) )