Theorem hbii1 2013

Description: Bound-variable hypothesis builder for indexed intersection.

Assertion

Ref	Expression
hbii1	⊢ (y ∈ ∩x ∈ A B → ∀x y ∈ ∩x ∈ A B)

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

Proof of Theorem hbii1

Step	Hyp	Ref	Expression
1		hbra1 1237	. . 3 ⊢ (∀x ∈ A z ∈ B → ∀x∀x ∈ A z ∈ B)
2	1	hbab 1096	. 2 ⊢ (y ∈ {z∣∀x ∈ A z ∈ B} → ∀x y ∈ {z∣∀x ∈ A z ∈ B})
3		df-iin 1997	. . 3 ⊢ ∩x ∈ A B = {z∣∀x ∈ A z ∈ B}
4	3	eleq2i 1153	. 2 ⊢ (y ∈ ∩x ∈ A B ↔ y ∈ {z∣∀x ∈ A z ∈ B})
5	4	bial 695	. 2 ⊢ (∀x y ∈ ∩x ∈ A B ↔ ∀x y ∈ {z∣∀x ∈ A z ∈ B})
6	2, 4, 5	3imtr4 192	1 ⊢ (y ∈ ∩x ∈ A B → ∀x y ∈ ∩x ∈ A B)

Syntax hints:   → wi 2  ∀wal 672  {cab 1090   ∈ wcel 1092  ∀wral 1201  ∩ciin 1995

This theorem is referenced by:  scott0 3542

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-iin 1997

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