Theorem rabid2

Description: An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003) (Proof shortened by Andrew Salmon, 30-May-2011)

	Ref	Expression
Assertion	rabid2	⊢ ( 𝐴 = { 𝑥 ∈ 𝐴 ∣ 𝜑 } ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		abeq2	⊢ ( 𝐴 = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
2		pm4.71	⊢ ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
3	2	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
4	1 3	bitr4i	⊢ ( 𝐴 = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
5		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
6	5	eqeq2i	⊢ ( 𝐴 = { 𝑥 ∈ 𝐴 ∣ 𝜑 } ↔ 𝐴 = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } )
7		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
8	4 6 7	3bitr4i	⊢ ( 𝐴 = { 𝑥 ∈ 𝐴 ∣ 𝜑 } ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )