Theorem elintrab 1977

Description: Membership in the intersection of a class abstraction.

Hypothesis

Ref	Expression
inteqab.1	⊢ A ∈ V

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem elintrab

Step	Hyp	Ref	Expression
1		inteqab.1	. . . 4 ⊢ A ∈ V
2	1	elintab 1976	. . 3 ⊢ (A ∈ ∩{x∣(x ∈ B ∧ φ)} ↔ ∀x((x ∈ B ∧ φ) → A ∈ x))
3		impexp 276	. . . 4 ⊢ (((x ∈ B ∧ φ) → A ∈ x) ↔ (x ∈ B → (φ → A ∈ x)))
4	3	bial 695	. . 3 ⊢ (∀x((x ∈ B ∧ φ) → A ∈ x) ↔ ∀x(x ∈ B → (φ → A ∈ x)))
5	2, 4	bitr 151	. 2 ⊢ (A ∈ ∩{x∣(x ∈ B ∧ φ)} ↔ ∀x(x ∈ B → (φ → A ∈ x)))
6		df-rab 1208	. . . 4 ⊢ {x ∈ B∣φ} = {x∣(x ∈ B ∧ φ)}
7	6	inteqi 1969	. . 3 ⊢ ∩{x ∈ B∣φ} = ∩{x∣(x ∈ B ∧ φ)}
8	7	eleq2i 1153	. 2 ⊢ (A ∈ ∩{x ∈ B∣φ} ↔ A ∈ ∩{x∣(x ∈ B ∧ φ)})
9		df-ral 1205	. 2 ⊢ (∀x ∈ B (φ → A ∈ x) ↔ ∀x(x ∈ B → (φ → A ∈ x)))
10	5, 8, 9	3bitr4 158	1 ⊢ (A ∈ ∩{x ∈ B∣φ} ↔ ∀x ∈ B (φ → A ∈ x))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  {cab 1090   ∈ wcel 1092  ∀wral 1201  {crab 1204  Vcvv 1348  ∩cint 1965

This theorem is referenced by:  intmin 1982  rankun 3535  elspan 5449

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-ral 1205  df-rab 1208  df-v 1349  df-int 1966

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