Theorem elintrab 4963

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)

Hypothesis

Ref	Expression
elintab.ex	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elintrab	⊢ (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem elintrab

Step	Hyp	Ref	Expression
1		elintab.ex	. . . 4 ⊢ 𝐴 ∈ V
2	1	elintab 4961	. . 3 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝐴 ∈ 𝑥))
3		impexp 450	. . . 4 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝐴 ∈ 𝑥) ↔ (𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ∈ 𝑥)))
4	3	albii 1814	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝐴 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ∈ 𝑥)))
5	2, 4	bitri 275	. 2 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ∈ 𝑥)))
6		df-rab 3430	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
7	6	inteqi 4953	. . 3 ⊢ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} = ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
8	7	eleq2i 2821	. 2 ⊢ (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
9		df-ral 3059	. 2 ⊢ (∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ∈ 𝑥)))
10	5, 8, 9	3bitr4i 303	1 ⊢ (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1532   ∈ wcel 2099  {cab 2705  ∀wral 3058  {crab 3429  Vcvv 3471  ∩ cint 4949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-int 4950

This theorem is referenced by:  elintrabg  4964  intmin  4971  naddelim  8707  rankunb  9874  isf34lem4  10401  ist1-3  23266  filufint  23837  elspani  31366  fldgensdrg  33014  evls1fldgencl  33358  ldsysgenld  33779  ldgenpisyslem1  33782  kur14lem9  34824  pclclN  39364  elpclN  39365  naddwordnexlem4  42831  lcosslsp  47506