Theorem rabid 1308

Description: An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16.

Assertion

Ref	Expression
rabid	⊢ (x ∈ {x ∈ A∣φ} ↔ (x ∈ A ∧ φ))

Proof of Theorem rabid

Step	Hyp	Ref	Expression
1		df-rab 1208	. 2 ⊢ {x ∈ A∣φ} = {x∣(x ∈ A ∧ φ)}
2	1	cleqabi 1176	1 ⊢ (x ∈ {x ∈ A∣φ} ↔ (x ∈ A ∧ φ))

Syntax hints:   ↔ wb 127   ∧ wa 196   ∈ wcel 1092  {crab 1204

This theorem is referenced by:  cleqrabi 1347  nnwos 4610

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rab 1208

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