Theorem hbint 1975

Description: Bound-variable hypothesis builder for intersection.

Hypothesis

Ref	Expression
hbint.1	⊢ (y ∈ A → ∀x y ∈ A)

Assertion

Ref	Expression
hbint	⊢ (y ∈ ∩A → ∀x y ∈ ∩A)

Distinct variable group(s):   y,A   x,y

Proof of Theorem hbint

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 ⊢ (y ∈ z → ∀x y ∈ z)
2		hbint.1	. . . . 5 ⊢ (y ∈ A → ∀x y ∈ A)
3	1, 2	hbel 1172	. . . 4 ⊢ (z ∈ A → ∀x z ∈ A)
4	3, 1	hbim 702	. . 3 ⊢ ((z ∈ A → y ∈ z) → ∀x(z ∈ A → y ∈ z))
5	4	hbal 700	. 2 ⊢ (∀z(z ∈ A → y ∈ z) → ∀x∀z(z ∈ A → y ∈ z))
6		visset 1350	. . 3 ⊢ y ∈ V
7	6	elint 1971	. 2 ⊢ (y ∈ ∩A ↔ ∀z(z ∈ A → y ∈ z))
8	7	bial 695	. 2 ⊢ (∀x y ∈ ∩A ↔ ∀x∀z(z ∈ A → y ∈ z))
9	5, 7, 8	3imtr4 192	1 ⊢ (y ∈ ∩A → ∀x y ∈ ∩A)

Syntax hints:   → wi 2  ∀wal 672   ∈ wel 803   ∈ wcel 1092  ∩cint 1965

This theorem is referenced by:  onminsb 2264  onminex 2275  oawordeulem 3156  unblem2 3432  unblem3 3433  tz9.12lem3 3505  rankid 3516  cardmin 3666  alephordlem1 3677  cardaleph 3690

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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