This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem nfiing

Description: Bound-variable hypothesis builder for indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 . See nfiin for a version with more disjoint variable conditions, but not requiring ax-13 . (Contributed by Mario Carneiro, 25-Jan-2014) (New usage is discouraged.)

Ref Expression
Hypotheses nfiung.1 ⊢ Ⅎ _ y A
nfiung.2 ⊢ Ⅎ _ y B
Assertion nfiing ⊢ Ⅎ _ y ⋂ x ∈ A B

Proof

Step Hyp Ref Expression
1 nfiung.1 ⊢ Ⅎ _ y A
2 nfiung.2 ⊢ Ⅎ _ y B
3 df-iin ⊢ ⋂ x ∈ A B = z | ∀ x ∈ A z ∈ B
4 2 nfcri ⊢ Ⅎ y z ∈ B
5 1 4 nfral ⊢ Ⅎ y ∀ x ∈ A z ∈ B
6 5 nfabg ⊢ Ⅎ _ y z | ∀ x ∈ A z ∈ B
7 3 6 nfcxfr ⊢ Ⅎ _ y ⋂ x ∈ A B