Theorem dfrab2 1696

Description: Alternate definition of restricted class abstraction.

Assertion

Ref	Expression
dfrab2	⊢ {x ∈ A∣φ} = ({x∣φ} ∩ A)

Distinct variable group(s):   x,A

Proof of Theorem dfrab2

Step	Hyp	Ref	Expression
1		df-rab 1208	. 2 ⊢ {x ∈ A∣φ} = {x∣(x ∈ A ∧ φ)}
2		inab 1692	. . 3 ⊢ ({x∣x ∈ A} ∩ {x∣φ}) = {x∣(x ∈ A ∧ φ)}
3		abid2 1186	. . . 4 ⊢ {x∣x ∈ A} = A
4	3	ineq1i 1641	. . 3 ⊢ ({x∣x ∈ A} ∩ {x∣φ}) = (A ∩ {x∣φ})
5	2, 4	eqtr3 1121	. 2 ⊢ {x∣(x ∈ A ∧ φ)} = (A ∩ {x∣φ})
6		incom 1636	. 2 ⊢ (A ∩ {x∣φ}) = ({x∣φ} ∩ A)
7	1, 5, 6	3eqtr 1123	1 ⊢ {x ∈ A∣φ} = ({x∣φ} ∩ A)

Syntax hints:   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  {crab 1204   ∩ cin 1486

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-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  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  df-v 1349  df-in 1491

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