Theorem nfiin 4986

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 4988 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 4957	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfiun.1	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfiun.2	. . . . 5 ⊢ Ⅎ𝑦𝐵
4	3	nfcri 2918	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5	2, 4	nfralw 3313	. . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfab 2932	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
7	1, 6	nfcxfr 2924	1 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵

This proof depends on syntax axioms:   ∈ wcel 2146  {cab 2741  Ⅎwnfc 2911  ∀wral 3080  ∩ ciin 4955

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-iin 4957

This theorem is used by:  iinab  5029  fnlimcnv  46819  fnlimfvre  46826  fnlimabslt  46831  iinhoiicc  47826  preimageiingt  47872  preimaleiinlt  47873  smflimlem6  47928  smflim  47929  smflim2  47958  smfsup  47966  smfsupmpt  47967  smfsupxr  47968  smfinflem  47969  smfinf  47970  smflimsup  47980  smfliminf  47983  fsupdm  47994  finfdm  47998