Theorem elintrab 4423

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

Hypothesis

Ref	Expression
inteqab.1	⊢ 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		inteqab.1	. . . 4 ⊢ 𝐴 ∈ V
2	1	elintab 4422	. . 3 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝐴 ∈ 𝑥))
3		impexp 461	. . . 4 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝐴 ∈ 𝑥) ↔ (𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ∈ 𝑥)))
4	3	albii 1737	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝐴 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ∈ 𝑥)))
5	2, 4	bitri 263	. 2 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ∈ 𝑥)))
6		df-rab 2905	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
7	6	inteqi 4414	. . 3 ⊢ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} = ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
8	7	eleq2i 2680	. 2 ⊢ (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
9		df-ral 2901	. 2 ⊢ (∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ∈ 𝑥)))
10	5, 8, 9	3bitr4i 291	1 ⊢ (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   ∈ wcel 1977  {cab 2596  ∀wral 2896  {crab 2900  Vcvv 3173  ∩ cint 4410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-int 4411

This theorem is referenced by:  elintrabg  4424  intmin  4432  rankunb  8596  isf34lem4  9082  ist1-3  20963  filufint  21534  elspani  27786  ldsysgenld  29550  ldgenpisyslem1  29553  kur14lem9  30450  pclclN  34195  elpclN  34196  lcosslsp  42021