Theorem nfiunya 3888

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

Hypotheses

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

Assertion

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

Distinct variable group:   x, A

Allowed substitution hints:   A( y)   B( x, y)

Proof of Theorem nfiunya

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 3862	. 2 |- U_ x e. A B = { z | E. x e. A z e. B }
2		nfiunya.1	. . . 4 |- F/_ y A
3		nfiunya.2	. . . . 5 |- F/_ y B
4	3	nfcri 2300	. . . 4 |- F/ y z e. B
5	2, 4	nfrexya 2505	. . 3 |- F/ y E. x e. A z e. B
6	5	nfab 2311	. 2 |- F/_ y { z | E. x e. A z e. B }
7	1, 6	nfcxfr 2303	1 |- F/_ y U_ x e. A B

Syntax hints:   e. wcel 2135   {cab 2150   F/_wnfc 2293   E.wrex 2443   U_ciun 3860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-iun 3862

This theorem is referenced by: (None)