Theorem nfiin 4983

Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2404. See nfiing 4985 for a less restrictive version requiring more axioms. (Revised by GG, 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 4953	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfiun.1	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfiun.2	. . . . 5 ⊢ Ⅎ𝑦𝐵
4	3	nfcri 2917	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5	2, 4	nfralw 3310	. . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfab 2931	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
7	1, 6	nfcxfr 2923	1 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2143  {cab 2741  Ⅎwnfc 2910  ∀wral 3077  ∩ ciin 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-nf 1805  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3078  df-iin 4953

This theorem is referenced by:  iinab  5026  fnlimcnv  46242  fnlimfvre  46249  fnlimabslt  46254  iinhoiicc  47249  preimageiingt  47295  preimaleiinlt  47296  smflimlem6  47351  smflim  47352  smflim2  47381  smfsup  47389  smfsupmpt  47390  smfsupxr  47391  smfinflem  47392  smfinf  47393  smflimsup  47403  smfliminf  47406  fsupdm  47417  finfdm  47421