Theorem nfii1 4964

Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)

Assertion

Ref	Expression
nfii1	⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵

Proof of Theorem nfii1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 4932	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2		nfra1 3144	. . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfab 2914	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
4	1, 3	nfcxfr 2906	1 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2109  {cab 2716  Ⅎwnfc 2888  ∀wral 3065  ∩ ciin 4930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-iin 4932

This theorem is referenced by:  dmiin  5859  scott0  9628  gruiin  10550  zarclsiin  31800  iinssiin  42631  iooiinicc  43034  iooiinioc  43048  fnlimfvre  43169  fnlimabslt  43174  meaiininclem  43978  hspdifhsp  44108  smflimlem2  44258  smflim  44263  smflimmpt  44294  smfsuplem1  44295  smfsupmpt  44299  smfsupxr  44300  smfinflem  44301  smfinfmpt  44303  smflimsuplem7  44310  smflimsuplem8  44311  smflimsupmpt  44313  smfliminfmpt  44316