Theorem rabab 1359

Description: A class abstraction restricted to the universe is unrestricted.

Assertion

Ref	Expression
rabab	⊢ {x ∈ V∣φ} = {x∣φ}

Proof of Theorem rabab

Step	Hyp	Ref	Expression
1		df-rab 1208	. 2 ⊢ {x ∈ V∣φ} = {x∣(x ∈ V ∧ φ)}
2		pm3.27 260	. . . 4 ⊢ ((x ∈ V ∧ φ) → φ)
3		visset 1350	. . . . 5 ⊢ x ∈ V
4	3	jctl 238	. . . 4 ⊢ (φ → (x ∈ V ∧ φ))
5	2, 4	impbi 139	. . 3 ⊢ ((x ∈ V ∧ φ) ↔ φ)
6	5	biabi 1181	. 2 ⊢ {x∣(x ∈ V ∧ φ)} = {x∣φ}
7	1, 6	eqtr 1119	1 ⊢ {x ∈ V∣φ} = {x∣φ}

Syntax hints:   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  {crab 1204  Vcvv 1348

This theorem is referenced by:  iunab 2023

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

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