Theorem elrabf 1421

Description: Membership in a restricted class abstraction with implicit substitution. This version has bound variable hypotheses in place of distinct variable restrictions.

Hypotheses

Ref	Expression
elrabf.1	⊢ (y ∈ A → ∀x y ∈ A)
elrabf.2	⊢ (y ∈ B → ∀x y ∈ B)
elrabf.3	⊢ (ψ → ∀xψ)
elrabf.4	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
elrabf	⊢ (A ∈ {x ∈ B∣φ} ↔ (A ∈ B ∧ ψ))

Distinct variable group(s):   x,y   y,A   y,B

Proof of Theorem elrabf

Step	Hyp	Ref	Expression
1		elisset 1354	. 2 ⊢ (A ∈ {x ∈ B∣φ} → A ∈ V)
2		elisset 1354	. . 3 ⊢ (A ∈ B → A ∈ V)
3	2	adantr 306	. 2 ⊢ ((A ∈ B ∧ ψ) → A ∈ V)
4		elrabf.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
5		elrabf.2	. . . . . 6 ⊢ (y ∈ B → ∀x y ∈ B)
6	4, 5	hbel 1172	. . . . 5 ⊢ (A ∈ B → ∀x A ∈ B)
7		elrabf.3	. . . . 5 ⊢ (ψ → ∀xψ)
8	6, 7	hban 704	. . . 4 ⊢ ((A ∈ B ∧ ψ) → ∀x(A ∈ B ∧ ψ))
9		eleq1 1149	. . . . 5 ⊢ (x = A → (x ∈ B ↔ A ∈ B))
10		elrabf.4	. . . . 5 ⊢ (x = A → (φ ↔ ψ))
11	9, 10	anbi12d 476	. . . 4 ⊢ (x = A → ((x ∈ B ∧ φ) ↔ (A ∈ B ∧ ψ)))
12	4, 8, 11	elabgf 1416	. . 3 ⊢ (A ∈ V → (A ∈ {x∣(x ∈ B ∧ φ)} ↔ (A ∈ B ∧ ψ)))
13		df-rab 1208	. . . 4 ⊢ {x ∈ B∣φ} = {x∣(x ∈ B ∧ φ)}
14	13	eleq2i 1153	. . 3 ⊢ (A ∈ {x ∈ B∣φ} ↔ A ∈ {x∣(x ∈ B ∧ φ)})
15	12, 14	syl5bb 410	. 2 ⊢ (A ∈ V → (A ∈ {x ∈ B∣φ} ↔ (A ∈ B ∧ ψ)))
16	1, 3, 15	pm5.21nii 504	1 ⊢ (A ∈ {x ∈ B∣φ} ↔ (A ∈ B ∧ ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  {cab 1090   = wceq 1091   ∈ wcel 1092  {crab 1204  Vcvv 1348

This theorem is referenced by:  elrab 1422  elrabsf 1456  onminsb 2264  tz9.12lem3 3505  ondomcard 3663

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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