Theorem nfunid 4443

Description: Deduction version of nfuni 4442. (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 4438	. 2 |- U. A = { y | E. z e. A y e. z }
2		nfv 1843	. . 3 |- F/ y ph
3		nfv 1843	. . . 4 |- F/ z ph
4		nfunid.3	. . . 4 |- ( ph -> F/_ x A )
5		nfvd 1844	. . . 4 |- ( ph -> F/ x y e. z )
6	3, 4, 5	nfrexd 3006	. . 3 |- ( ph -> F/ x E. z e. A y e. z )
7	2, 6	nfabd 2785	. 2 |- ( ph -> F/_ x { y | E. z e. A y e. z } )
8	1, 7	nfcxfrd 2763	1 |- ( ph -> F/_ x U. A )

Syntax hints:   -> wi 4   {cab 2608   F/_wnfc 2751   E.wrex 2913   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-uni 4437

This theorem is referenced by:  dfnfc2  4454  dfnfc2OLD  4455  nfiotad  5854