Theorem nfii1 4977

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 4942	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2		nfra1 3256	. . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfab 2900	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
4	1, 3	nfcxfr 2892	1 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2111  {cab 2709  Ⅎwnfc 2879  ∀wral 3047  ∩ ciin 4940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-iin 4942

This theorem is referenced by:  dmiin  5892  scott0  9779  gruiin  10701  zarclsiin  33884  iinssiin  45236  iooiinicc  45652  iooiinioc  45666  fnlimfvre  45782  fnlimabslt  45787  meaiininclem  46594  hspdifhsp  46724  smflimlem2  46880  smflim  46885  smflimmpt  46918  smfsuplem1  46919  smfsupmpt  46923  smfsupxr  46924  smfinflem  46925  smfinfmpt  46927  smflimsuplem7  46934  smflimsuplem8  46935  smflimsupmpt  46937  smfliminfmpt  46940  fsupdm  46950  finfdm  46954  iinfssc  49168  iinfsubc  49169