Theorem nfiotad 5854

Description: Deduction version of nfiota 5855. (Contributed by NM, 18-Feb-2013.)

Hypotheses

Ref	Expression
nfiotad.1	|- F/ y ph
nfiotad.2	|- ( ph -> F/ x ps )

Assertion

Ref	Expression
nfiotad	|- ( ph -> F/_ x ( iota y ps ) )

Proof of Theorem nfiotad

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiota2 5852	. 2 |- ( iota y ps ) = U. { z | A. y ( ps <-> y = z ) }
2		nfv 1843	. . . 4 |- F/ z ph
3		nfiotad.1	. . . . 5 |- F/ y ph
4		nfiotad.2	. . . . . . 7 |- ( ph -> F/ x ps )
5	4	adantr 481	. . . . . 6 |- ( ( ph /\ -. A. x x = y ) -> F/ x ps )
6		nfeqf1 2299	. . . . . . 7 |- ( -. A. x x = y -> F/ x y = z )
7	6	adantl 482	. . . . . 6 |- ( ( ph /\ -. A. x x = y ) -> F/ x y = z )
8	5, 7	nfbid 1832	. . . . 5 |- ( ( ph /\ -. A. x x = y ) -> F/ x ( ps <-> y = z ) )
9	3, 8	nfald2 2331	. . . 4 |- ( ph -> F/ x A. y ( ps <-> y = z ) )
10	2, 9	nfabd 2785	. . 3 |- ( ph -> F/_ x { z | A. y ( ps <-> y = z ) } )
11	10	nfunid 4443	. 2 |- ( ph -> F/_ x U. { z | A. y ( ps <-> y = z ) } )
12	1, 11	nfcxfrd 2763	1 |- ( ph -> F/_ x ( iota y ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   F/wnf 1708   {cab 2608   F/_wnfc 2751   U.cuni 4436   iotacio 5849

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-sn 4178  df-uni 4437  df-iota 5851

This theorem is referenced by:  nfiota  5855  nfriotad  6619