Theorem elintab 1976

Description: Membership in the intersection of a class abstraction.

Hypothesis

Ref	Expression
inteqab.1	⊢ A ∈ V

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem elintab

Step	Hyp	Ref	Expression
1		inteqab.1	. . 3 ⊢ A ∈ V
2	1	elint 1971	. 2 ⊢ (A ∈ ∩{x∣φ} ↔ ∀y(y ∈ {x∣φ} → A ∈ y))
3		hbab1 1095	. . . 4 ⊢ (y ∈ {x∣φ} → ∀x y ∈ {x∣φ})
4		ax-17 925	. . . 4 ⊢ (A ∈ y → ∀x A ∈ y)
5	3, 4	hbim 702	. . 3 ⊢ ((y ∈ {x∣φ} → A ∈ y) → ∀x(y ∈ {x∣φ} → A ∈ y))
6		ax-17 925	. . 3 ⊢ ((φ → A ∈ x) → ∀y(φ → A ∈ x))
7		eleq1 1149	. . . . 5 ⊢ (y = x → (y ∈ {x∣φ} ↔ x ∈ {x∣φ}))
8		abid 1094	. . . . 5 ⊢ (x ∈ {x∣φ} ↔ φ)
9	7, 8	syl6bb 414	. . . 4 ⊢ (y = x → (y ∈ {x∣φ} ↔ φ))
10		eleq2 1150	. . . 4 ⊢ (y = x → (A ∈ y ↔ A ∈ x))
11	9, 10	imbi12d 474	. . 3 ⊢ (y = x → ((y ∈ {x∣φ} → A ∈ y) ↔ (φ → A ∈ x)))
12	5, 6, 11	cbval 848	. 2 ⊢ (∀y(y ∈ {x∣φ} → A ∈ y) ↔ ∀x(φ → A ∈ x))
13	2, 12	bitr 151	1 ⊢ (A ∈ ∩{x∣φ} ↔ ∀x(φ → A ∈ x))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   = weq 797  {cab 1090   ∈ wcel 1092  Vcvv 1348  ∩cint 1965

This theorem is referenced by:  elintrab 1977  dfom3 3477  1nn 4432  peano2nn 4433

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-int 1966

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