Theorem elintab 3790

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)

Hypothesis

Ref	Expression
inteqab.1	⊢ 𝐴 ∈ V

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elintab

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inteqab.1	. . 3 ⊢ 𝐴 ∈ V
2	1	elint 3785	. 2 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦))
3		nfsab1 2130	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
4		nfv 1509	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦
5	3, 4	nfim 1552	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦)
6		nfv 1509	. . 3 ⊢ Ⅎ𝑦(𝜑 → 𝐴 ∈ 𝑥)
7		eleq1 2203	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
8		abid 2128	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
9	7, 8	syl6bb 195	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))
10		eleq2 2204	. . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥))
11	9, 10	imbi12d 233	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) ↔ (𝜑 → 𝐴 ∈ 𝑥)))
12	5, 6, 11	cbval 1728	. 2 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))
13	2, 12	bitri 183	1 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1330   ∈ wcel 1481  {cab 2126  Vcvv 2689  ∩ cint 3779

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-int 3780

This theorem is referenced by:  elintrab  3791  intmin4  3807  intab  3808  intid  4154