Theorem elrabf 2880

Description: Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)

Hypotheses

Ref	Expression
elrabf.1	⊢ Ⅎ𝑥𝐴
elrabf.2	⊢ Ⅎ𝑥𝐵
elrabf.3	⊢ Ⅎ𝑥𝜓
elrabf.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

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

Proof of Theorem elrabf

Step	Hyp	Ref	Expression
1		elex 2737	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} → 𝐴 ∈ V)
2		elex 2737	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
3	2	adantr 274	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → 𝐴 ∈ V)
4		df-rab 2453	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
5	4	eleq2i 2233	. . 3 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
6		elrabf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
7		elrabf.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfel 2317	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
9		elrabf.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
10	8, 9	nfan 1553	. . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓)
11		eleq1 2229	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
12		elrabf.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
13	11, 12	anbi12d 465	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)))
14	6, 10, 13	elabgf 2868	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)))
15	5, 14	syl5bb 191	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)))
16	1, 3, 15	pm5.21nii 694	1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  Ⅎwnf 1448   ∈ wcel 2136  {cab 2151  Ⅎwnfc 2295  {crab 2448  Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728

This theorem is referenced by:  elrab  2882  frind  4330  rabxfrd  4447  infssuzcldc  11884  nnwosdc  11972