Theorem elabs2 1457

Description: Membership in a class abstraction, expressed in terms of class substitution. Theorem 6.13 of [Quine] p. 44.

Assertion

Ref	Expression
elabs2	⊢ (A ∈ {x∣φ} ↔ (A ∈ V ∧ [A / x]φ))

Proof of Theorem elabs2

Step	Hyp	Ref	Expression
1		df-rab 1208	. . . 4 ⊢ {x ∈ V∣φ} = {x∣(x ∈ V ∧ φ)}
2		visset 1350	. . . . . 6 ⊢ x ∈ V
3	2	biantrur 544	. . . . 5 ⊢ (φ ↔ (x ∈ V ∧ φ))
4	3	biabi 1181	. . . 4 ⊢ {x∣φ} = {x∣(x ∈ V ∧ φ)}
5	1, 4	eqtr4 1122	. . 3 ⊢ {x ∈ V∣φ} = {x∣φ}
6	5	eleq2i 1153	. 2 ⊢ (A ∈ {x ∈ V∣φ} ↔ A ∈ {x∣φ})
7		ax-17 925	. . 3 ⊢ (y ∈ V → ∀x y ∈ V)
8	7	elrabsf 1456	. 2 ⊢ (A ∈ {x ∈ V∣φ} ↔ (A ∈ V ∧ [A / x]φ))
9	6, 8	bitr3 153	1 ⊢ (A ∈ {x∣φ} ↔ (A ∈ V ∧ [A / x]φ))

Syntax hints:   ↔ wb 127   ∧ wa 196  {cab 1090   ∈ wcel 1092  {crab 1204  Vcvv 1348  [wsbc 1440

This theorem is referenced by:  elabs 1458

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-or 197  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-sbc 1441

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