Theorem nfiunya 3957

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 3931	. 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 2343	. . . 4 |- F/ y z e. B
5	2, 4	nfrexya 2548	. . 3 |- F/ y E. x e. A z e. B
6	5	nfab 2354	. 2 |- F/_ y { z | E. x e. A z e. B }
7	1, 6	nfcxfr 2346	1 |- F/_ y U_ x e. A B

Syntax hints:   e. wcel 2177   {cab 2192   F/_wnfc 2336   E.wrex 2486   U_ciun 3929

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-iun 3931

This theorem is referenced by: (None)