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Metamath Proof Explorer

Theorem rabid2f

Description: An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003) (Proof shortened by Andrew Salmon, 30-May-2011) (Revised by Thierry Arnoux, 13-Mar-2017)

		Ref	Expression
	Hypothesis	rabid2f.1	⊢ Ⅎ 𝑥 𝐴
	Assertion	rabid2f	⊢ ( 𝐴 = { 𝑥 ∈ 𝐴 ∣ 𝜑 } ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		rabid2f.1	⊢ Ⅎ 𝑥 𝐴
2	1	eqabf	⊢ ( 𝐴 = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
3		pm4.71	⊢ ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
4	3	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
5	2 4	bitr4i	⊢ ( 𝐴 = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
6		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
7	6	eqeq2i	⊢ ( 𝐴 = { 𝑥 ∈ 𝐴 ∣ 𝜑 } ↔ 𝐴 = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } )
8		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
9	5 7 8	3bitr4i	⊢ ( 𝐴 = { 𝑥 ∈ 𝐴 ∣ 𝜑 } ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )