Theorem nfiunya 3944

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 3918	. 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 2333	. . . 4 |- F/ y z e. B
5	2, 4	nfrexya 2538	. . 3 |- F/ y E. x e. A z e. B
6	5	nfab 2344	. 2 |- F/_ y { z | E. x e. A z e. B }
7	1, 6	nfcxfr 2336	1 |- F/_ y U_ x e. A B

Syntax hints:   e. wcel 2167   {cab 2182   F/_wnfc 2326   E.wrex 2476   U_ciun 3916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-iun 3918

This theorem is referenced by: (None)