Theorem nfunid 3803

Description: Deduction version of nfuni 3802. (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 3798	. 2 |- U. A = { y | E. z e. A y e. z }
2		nfv 1521	. . 3 |- F/ y ph
3		nfv 1521	. . . 4 |- F/ z ph
4		nfunid.3	. . . 4 |- ( ph -> F/_ x A )
5		nfvd 1522	. . . 4 |- ( ph -> F/ x y e. z )
6	3, 4, 5	nfrexdxy 2504	. . 3 |- ( ph -> F/ x E. z e. A y e. z )
7	2, 6	nfabd 2332	. 2 |- ( ph -> F/_ x { y | E. z e. A y e. z } )
8	1, 7	nfcxfrd 2310	1 |- ( ph -> F/_ x U. A )

Syntax hints:   -> wi 4   {cab 2156   F/_wnfc 2299   E.wrex 2449   U.cuni 3796

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-uni 3797

This theorem is referenced by:  dfnfc2  3814  nfiotadw  5163