Theorem elintab 3938

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 3933	. 2 ⊢ (A ∈ ∩{x ∣ φ} ↔ ∀y(y ∈ {x ∣ φ} → A ∈ y))
3		nfsab1 2343	. . . 4 ⊢ Ⅎx y ∈ {x ∣ φ}
4		nfv 1619	. . . 4 ⊢ Ⅎx A ∈ y
5	3, 4	nfim 1813	. . 3 ⊢ Ⅎx(y ∈ {x ∣ φ} → A ∈ y)
6		nfv 1619	. . 3 ⊢ Ⅎy(φ → A ∈ x)
7		eleq1 2413	. . . . 5 ⊢ (y = x → (y ∈ {x ∣ φ} ↔ x ∈ {x ∣ φ}))
8		abid 2341	. . . . 5 ⊢ (x ∈ {x ∣ φ} ↔ φ)
9	7, 8	syl6bb 252	. . . 4 ⊢ (y = x → (y ∈ {x ∣ φ} ↔ φ))
10		eleq2 2414	. . . 4 ⊢ (y = x → (A ∈ y ↔ A ∈ x))
11	9, 10	imbi12d 311	. . 3 ⊢ (y = x → ((y ∈ {x ∣ φ} → A ∈ y) ↔ (φ → A ∈ x)))
12	5, 6, 11	cbval 1984	. 2 ⊢ (∀y(y ∈ {x ∣ φ} → A ∈ y) ↔ ∀x(φ → A ∈ x))
13	2, 12	bitri 240	1 ⊢ (A ∈ ∩{x ∣ φ} ↔ ∀x(φ → A ∈ x))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2860  ∩cint 3927

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-int 3928

This theorem is used by:  elintrab  3939  intmin4  3956  intab  3957  peano1  4403  peano2  4404  ncvspfin  4539  spfinsfincl  4540  clos1conn  5880  nchoicelem10  6299