Theorem nfiin 4953

Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2389. See nfiing 4955 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)

Hypotheses

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

Assertion

Ref	Expression
nfiin	⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfiin

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 4925	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfiun.1	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfiun.2	. . . . 5 ⊢ Ⅎ𝑦𝐵
4	3	nfcri 2974	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5	2, 4	nfralw 3228	. . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfab 2987	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
7	1, 6	nfcxfr 2978	1 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2113  {cab 2802  Ⅎwnfc 2964  ∀wral 3141  ∩ ciin 4923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-iin 4925

This theorem is referenced by:  iinab  4993  fnlimcnv  41954  fnlimfvre  41961  fnlimabslt  41966  iinhoiicc  42963  preimageiingt  43005  preimaleiinlt  43006  smflimlem6  43059  smflim  43060  smflim2  43087  smfsup  43095  smfsupmpt  43096  smfsupxr  43097  smfinflem  43098  smfinf  43099  smfinfmpt  43100  smflimsup  43109  smfliminf  43112