Theorem nfii1 3999

Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)

Assertion

Ref	Expression
nfii1	⊢ Ⅎx∩x ∈ A B

Proof of Theorem nfii1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 3973	. 2 ⊢ ∩x ∈ A B = {y ∣ ∀x ∈ A y ∈ B}
2		nfra1 2665	. . 3 ⊢ Ⅎx∀x ∈ A y ∈ B
3	2	nfab 2494	. 2 ⊢ Ⅎx{y ∣ ∀x ∈ A y ∈ B}
4	1, 3	nfcxfr 2487	1 ⊢ Ⅎx∩x ∈ A B

This proof depends on syntax axioms:   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477  ∀wral 2615  ∩ciin 3971

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-iin 3973

This theorem is used by:  dmiin  4966