Theorem elintab 3653

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 3648	. 2 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦))
3		nfsab1 2046	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
4		nfv 1437	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦
5	3, 4	nfim 1480	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦)
6		nfv 1437	. . 3 ⊢ Ⅎ𝑦(𝜑 → 𝐴 ∈ 𝑥)
7		eleq1 2116	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
8		abid 2044	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
9	7, 8	syl6bb 189	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))
10		eleq2 2117	. . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥))
11	9, 10	imbi12d 227	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) ↔ (𝜑 → 𝐴 ∈ 𝑥)))
12	5, 6, 11	cbval 1653	. 2 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))
13	2, 12	bitri 177	1 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))

Syntax hints:   → wi 4   ↔ wb 102  ∀wal 1257   ∈ wcel 1409  {cab 2042  Vcvv 2574  ∩ cint 3642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038

This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-int 3643

This theorem is referenced by:  elintrab  3654  intmin4  3670  intab  3671  intid  3987