Theorem nfunid 3587

Description: Deduction version of nfuni 3586. (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 3582	. 2 |- U. A = { y | E. z e. A y e. z }
2		nfv 1421	. . 3 |- F/ y ph
3		nfv 1421	. . . 4 |- F/ z ph
4		nfunid.3	. . . 4 |- ( ph -> F/_ x A )
5		nfvd 1422	. . . 4 |- ( ph -> F/ x y e. z )
6	3, 4, 5	nfrexdxy 2357	. . 3 |- ( ph -> F/ x E. z e. A y e. z )
7	2, 6	nfabd 2196	. 2 |- ( ph -> F/_ x { y | E. z e. A y e. z } )
8	1, 7	nfcxfrd 2176	1 |- ( ph -> F/_ x U. A )

Syntax hints:   -> wi 4   {cab 2026   F/_wnfc 2165   E.wrex 2307   U.cuni 3580

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-uni 3581

This theorem is referenced by:  dfnfc2  3598  nfiotadxy  4870