Theorem nfiing 4957

Description: Bound-variable hypothesis builder for indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2372. See nfiin 4955 for a version with more disjoint variable conditions, but not requiring ax-13 2372. (Contributed by Mario Carneiro, 25-Jan-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfiung.1	⊢ Ⅎ𝑦𝐴
nfiung.2	⊢ Ⅎ𝑦𝐵

Assertion

Ref	Expression
nfiing	⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵

Proof of Theorem nfiing

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 4927	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfiung.1	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfiung.2	. . . . 5 ⊢ Ⅎ𝑦𝐵
4	3	nfcri 2894	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5	2, 4	nfral 3153	. . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfabg 2914	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
7	1, 6	nfcxfr 2905	1 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2106  {cab 2715  Ⅎwnfc 2887  ∀wral 3064  ∩ ciin 4925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-iin 4927

This theorem is referenced by: (None)