Theorem cleqrabi 1347

Description: Inference rule from equality of a class variable and a restricted class abstraction.

Hypothesis

Ref	Expression
cleqrabi.1	⊢ A = {x ∈ B∣φ}

Assertion

Ref	Expression
cleqrabi	⊢ (x ∈ A ↔ (x ∈ B ∧ φ))

Proof of Theorem cleqrabi

Step	Hyp	Ref	Expression
1		cleqrabi.1	. . 3 ⊢ A = {x ∈ B∣φ}
2	1	eleq2i 1153	. 2 ⊢ (x ∈ A ↔ x ∈ {x ∈ B∣φ})
3		rabid 1308	. 2 ⊢ (x ∈ {x ∈ B∣φ} ↔ (x ∈ B ∧ φ))
4	2, 3	bitr 151	1 ⊢ (x ∈ A ↔ (x ∈ B ∧ φ))

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  {crab 1204

This theorem is referenced by:  tfis 2245

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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