Theorem nfunid 3751

Description: Deduction version of nfuni 3750. (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 3746	. 2 |- U. A = { y | E. z e. A y e. z }
2		nfv 1509	. . 3 |- F/ y ph
3		nfv 1509	. . . 4 |- F/ z ph
4		nfunid.3	. . . 4 |- ( ph -> F/_ x A )
5		nfvd 1510	. . . 4 |- ( ph -> F/ x y e. z )
6	3, 4, 5	nfrexdxy 2471	. . 3 |- ( ph -> F/ x E. z e. A y e. z )
7	2, 6	nfabd 2301	. 2 |- ( ph -> F/_ x { y | E. z e. A y e. z } )
8	1, 7	nfcxfrd 2280	1 |- ( ph -> F/_ x U. A )

Syntax hints:   -> wi 4   {cab 2126   F/_wnfc 2269   E.wrex 2418   U.cuni 3744

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-uni 3745

This theorem is referenced by:  dfnfc2  3762  nfiotadw  5099