Theorem nfiund 42421

Description: Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.)

Hypotheses

Ref	Expression
nfiund.1	|- F/ x ph
nfiund.2	|- ( ph -> F/_ y A )
nfiund.3	|- ( ph -> F/_ y B )

Assertion

Ref	Expression
nfiund	|- ( ph -> F/_ y U_ x e. A B )

Proof of Theorem nfiund

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		nfv 1843	. . 3 |- F/ z ph
3		nfiund.1	. . . 4 |- F/ x ph
4		nfiund.2	. . . 4 |- ( ph -> F/_ y A )
5		nfiund.3	. . . . 5 |- ( ph -> F/_ y B )
6	5	nfcrd 2771	. . . 4 |- ( ph -> F/ y z e. B )
7	3, 4, 6	nfrexd 3006	. . 3 |- ( ph -> F/ y E. x e. A z e. B )
8	2, 7	nfabd 2785	. 2 |- ( ph -> F/_ y { z | E. x e. A z e. B } )
9	1, 8	nfcxfrd 2763	1 |- ( ph -> F/_ y U_ x e. A B )

Syntax hints:   -> wi 4   F/wnf 1708   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: (None)