Theorem elrabf 3642

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 3471	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} → 𝐴 ∈ V)
2		elex 3471	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
3	2	adantr 486	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → 𝐴 ∈ V)
4		df-rab 3413	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
5	4	eleq2i 2852	. . 3 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
6		elrabf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
7		elrabf.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfel 2936	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
9		elrabf.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
10	8, 9	nfan 1932	. . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓)
11		eleq1 2848	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
12		elrabf.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
13	11, 12	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)))
14	6, 10, 13	elabgf 3628	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)))
15	5, 14	bitrid 286	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)))
16	1, 3, 15	pm5.21nii 381	1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2145  {cab 2738  Ⅎwnfc 2907  {crab 3412  Vcvv 3450

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rab 3413  df-v 3452

This theorem is used by:  rabtru  3643  invdisjrab  5090  rabxfrd  5382  f1ossf1o  7122  onminsb  7793  nnawordex  8625  tskwe  9955  rabssnn0fi  14050  iundisj  25776  ltsval2  27892  iundisjf  33062  iundisjfi  33267  bnj1388  35542  phpreu  38358  poimirlem26  38395  sticksstones1  43012  rfcnpre3  45867  rfcnpre4  45868  uzwo4  45887  disjinfi  46024  allbutfiinf  46248  fsumiunss  46405  fnlimfvre  46502  stoweidlem26  46854  stoweidlem27  46855  stoweidlem31  46859  stoweidlem34  46862  stoweidlem51  46879  stoweidlem52  46880  stoweidlem59  46887  fourierdlem20  46955  fourierdlem79  47013  pimdecfgtioc  47543  smfpimcclem  47635  prmdvdsfmtnof1lem1  48487