Theorem nfiotadxy 4890

Description: Deduction version of nfiotaxy 4891. (Contributed by Jim Kingdon, 21-Dec-2018.)

Hypotheses

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

Assertion

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

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem nfiotadxy

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiota2 4888	. 2 |- ( iota y ps ) = U. { z | A. y ( ps <-> y = z ) }
2		nfv 1461	. . . 4 |- F/ z ph
3		nfiotadxy.1	. . . . 5 |- F/ y ph
4		nfiotadxy.2	. . . . . 6 |- ( ph -> F/ x ps )
5		nfcv 2219	. . . . . . . 8 |- F/_ x y
6		nfcv 2219	. . . . . . . 8 |- F/_ x z
7	5, 6	nfeq 2226	. . . . . . 7 |- F/ x y = z
8	7	a1i 9	. . . . . 6 |- ( ph -> F/ x y = z )
9	4, 8	nfbid 1520	. . . . 5 |- ( ph -> F/ x ( ps <-> y = z ) )
10	3, 9	nfald 1683	. . . 4 |- ( ph -> F/ x A. y ( ps <-> y = z ) )
11	2, 10	nfabd 2237	. . 3 |- ( ph -> F/_ x { z | A. y ( ps <-> y = z ) } )
12	11	nfunid 3608	. 2 |- ( ph -> F/_ x U. { z | A. y ( ps <-> y = z ) } )
13	1, 12	nfcxfrd 2217	1 |- ( ph -> F/_ x ( iota y ps ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   = wceq 1284   F/wnf 1389   {cab 2067   F/_wnfc 2206   U.cuni 3601   iotacio 4885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-sn 3404  df-uni 3602  df-iota 4887

This theorem is referenced by:  nfiotaxy  4891  nfriotadxy  5496