Theorem nfii1 4997

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 4963	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2		nfra1 3295	. . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfab 2937	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
4	1, 3	nfcxfr 2929	1 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2149  {cab 2747  Ⅎwnfc 2916  ∀wral 3085  ∩ ciin 4961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-iin 4963

This theorem is referenced by:  dmiin  5944  scott0  9860  gruiin  10795  zarclsiin  34206  iinssiin  45773  iooiinicc  46184  iooiinioc  46198  fnlimfvre  46314  fnlimabslt  46319  meaiininclem  47126  hspdifhsp  47256  smflimlem2  47412  smflim  47417  smflimmpt  47450  smfsuplem1  47451  smfsupmpt  47455  smfsupxr  47456  smfinflem  47457  smfinfmpt  47459  smflimsuplem7  47466  smflimsuplem8  47467  smflimsupmpt  47469  smfliminfmpt  47472  fsupdm  47482  finfdm  47486  iinfssc  49754  iinfsubc  49755