Theorem nfunid 3818

Description: Deduction version of nfuni 3817. (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 3813	. 2 |- U. A = { y | E. z e. A y e. z }
2		nfv 1528	. . 3 |- F/ y ph
3		nfv 1528	. . . 4 |- F/ z ph
4		nfunid.3	. . . 4 |- ( ph -> F/_ x A )
5		nfvd 1529	. . . 4 |- ( ph -> F/ x y e. z )
6	3, 4, 5	nfrexdxy 2511	. . 3 |- ( ph -> F/ x E. z e. A y e. z )
7	2, 6	nfabd 2339	. 2 |- ( ph -> F/_ x { y | E. z e. A y e. z } )
8	1, 7	nfcxfrd 2317	1 |- ( ph -> F/_ x U. A )

Syntax hints:   -> wi 4   {cab 2163   F/_wnfc 2306   E.wrex 2456   U.cuni 3811

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-uni 3812

This theorem is referenced by:  dfnfc2  3829  nfiotadw  5183