Theorem nfunid 3895

Description: Deduction version of nfuni 3894. (Contributed by NM, 18-Feb-2013.)

Hypothesis

Ref	Expression
nfunid.3	|- ( ph -> F/_ x A )

Assertion

Ref	Expression
nfunid	|- ( ph -> F/_ x U. A )

Proof of Theorem nfunid

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfuni2 3890	. 2 |- U. A = { y | E. z e. A y e. z }
2		nfv 1574	. . 3 |- F/ y ph
3		nfv 1574	. . . 4 |- F/ z ph
4		nfunid.3	. . . 4 |- ( ph -> F/_ x A )
5		nfvd 1575	. . . 4 |- ( ph -> F/ x y e. z )
6	3, 4, 5	nfrexdxy 2564	. . 3 |- ( ph -> F/ x E. z e. A y e. z )
7	2, 6	nfabd 2392	. 2 |- ( ph -> F/_ x { y | E. z e. A y e. z } )
8	1, 7	nfcxfrd 2370	1 |- ( ph -> F/_ x U. A )

Syntax hints:   -> wi 4   {cab 2215   F/_wnfc 2359   E.wrex 2509   U.cuni 3888

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-uni 3889

This theorem is referenced by:  dfnfc2  3906  nfiotadw  5281