Theorem nfiing 4985

Description: Bound-variable hypothesis builder for indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2402. See nfiin 4983 for a version with more disjoint variable conditions, but not requiring ax-13 2402. (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 4954	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfiung.1	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfiung.2	. . . . 5 ⊢ Ⅎ𝑦𝐵
4	3	nfcri 2915	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5	2, 4	nfral 3360	. . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfabg 2930	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
7	1, 6	nfcxfr 2921	1 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵

This proof depends on syntax axioms:   ∈ wcel 2145  {cab 2739  Ⅎwnfc 2908  ∀wral 3077  ∩ ciin 4952

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-13 2402  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-iin 4954

This theorem is used by: (None)