Theorem nfunid 3857

Description: Deduction version of nfuni 3856. (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 3852	. 2 |- U. A = { y | E. z e. A y e. z }
2		nfv 1551	. . 3 |- F/ y ph
3		nfv 1551	. . . 4 |- F/ z ph
4		nfunid.3	. . . 4 |- ( ph -> F/_ x A )
5		nfvd 1552	. . . 4 |- ( ph -> F/ x y e. z )
6	3, 4, 5	nfrexdxy 2540	. . 3 |- ( ph -> F/ x E. z e. A y e. z )
7	2, 6	nfabd 2368	. 2 |- ( ph -> F/_ x { y | E. z e. A y e. z } )
8	1, 7	nfcxfrd 2346	1 |- ( ph -> F/_ x U. A )

Syntax hints:   -> wi 4   {cab 2191   F/_wnfc 2335   E.wrex 2485   U.cuni 3850

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-uni 3851

This theorem is referenced by:  dfnfc2  3868  nfiotadw  5235