Theorem nfunid 3796

Description: Deduction version of nfuni 3795. (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 3791	. 2 |- U. A = { y | E. z e. A y e. z }
2		nfv 1516	. . 3 |- F/ y ph
3		nfv 1516	. . . 4 |- F/ z ph
4		nfunid.3	. . . 4 |- ( ph -> F/_ x A )
5		nfvd 1517	. . . 4 |- ( ph -> F/ x y e. z )
6	3, 4, 5	nfrexdxy 2500	. . 3 |- ( ph -> F/ x E. z e. A y e. z )
7	2, 6	nfabd 2328	. 2 |- ( ph -> F/_ x { y | E. z e. A y e. z } )
8	1, 7	nfcxfrd 2306	1 |- ( ph -> F/_ x U. A )

Syntax hints:   -> wi 4   {cab 2151   F/_wnfc 2295   E.wrex 2445   U.cuni 3789

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-uni 3790

This theorem is referenced by:  dfnfc2  3807  nfiotadw  5156