Theorem nfiin 4549

Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)

Hypotheses

Ref	Expression
nfiun.1	|- F/_ y A
nfiun.2	|- F/_ y B

Assertion

Ref	Expression
nfiin	|- F/_ y |^|_ x e. A B

Proof of Theorem nfiin

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 4523	. 2 |- |^|_ x e. A B = { z | A. x e. A z e. B }
2		nfiun.1	. . . 4 |- F/_ y A
3		nfiun.2	. . . . 5 |- F/_ y B
4	3	nfcri 2758	. . . 4 |- F/ y z e. B
5	2, 4	nfral 2945	. . 3 |- F/ y A. x e. A z e. B
6	5	nfab 2769	. 2 |- F/_ y { z | A. x e. A z e. B }
7	1, 6	nfcxfr 2762	1 |- F/_ y |^|_ x e. A B

Syntax hints:   e. wcel 1990   {cab 2608   F/_wnfc 2751   A.wral 2912   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-iin 4523

This theorem is referenced by:  iinab  4581  fnlimcnv  39899  fnlimfvre  39906  fnlimabslt  39911  iinhoiicc  40888  preimageiingt  40930  preimaleiinlt  40931  smflimlem6  40984  smflim  40985  smflim2  41012  smfsup  41020  smfsupmpt  41021  smfsupxr  41022  smfinflem  41023  smfinf  41024  smfinfmpt  41025  smflimsup  41034  smfliminf  41037