Theorem nfii1 4994

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 4962	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2		nfra1 3265	. . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfab 2908	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
4	1, 3	nfcxfr 2900	1 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2106  {cab 2708  Ⅎwnfc 2882  ∀wral 3060  ∩ ciin 4960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-iin 4962

This theorem is referenced by:  dmiin  5913  scott0  9831  gruiin  10755  zarclsiin  32541  iinssiin  43461  iooiinicc  43900  iooiinioc  43914  fnlimfvre  44035  fnlimabslt  44040  meaiininclem  44847  hspdifhsp  44977  smflimlem2  45133  smflim  45138  smflimmpt  45171  smfsuplem1  45172  smfsupmpt  45176  smfsupxr  45177  smfinflem  45178  smfinfmpt  45180  smflimsuplem7  45187  smflimsuplem8  45188  smflimsupmpt  45190  smfliminfmpt  45193  fsupdm  45203  finfdm  45207