Theorem nfiun 4548

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

Hypotheses

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

Assertion

Ref	Expression
nfiun	|- F/_ y U_ x e. A B

Proof of Theorem nfiun

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 4522	. 2 |- U_ x e. A B = { z | E. 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	nfrex 3007	. . 3 |- F/ y E. x e. A z e. B
6	5	nfab 2769	. 2 |- F/_ y { z | E. x e. A z e. B }
7	1, 6	nfcxfr 2762	1 |- F/_ y U_ x e. A B

Syntax hints:   e. wcel 1990   {cab 2608   F/_wnfc 2751   E.wrex 2913   U_ciun 4520

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-rex 2918  df-iun 4522

This theorem is referenced by:  iunab  4566  disjxiun  4649  disjxiunOLD  4650  ovoliunnul  23275  iundisjf  29402  iundisj2f  29403  iundisjfi  29555  iundisj2fi  29556  bnj1498  31129  trpredlem1  31727  trpredrec  31738  ss2iundf  37951  fnlimcnv  39899  fnlimfvre  39906  fnlimabslt  39911  smfaddlem1  40971  smflimlem6  40984  smflim  40985  smfmullem4  41001  smflim2  41012  smflimsup  41034  smfliminf  41037