Theorem nfiotadxy 4870

Description: Deduction version of nfiotaxy 4871. (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 4868	. 2 |- ( iota y ps ) = U. { z | A. y ( ps <-> y = z ) }
2		nfv 1421	. . . 4 |- F/ z ph
3		nfiotadxy.1	. . . . 5 |- F/ y ph
4		nfiotadxy.2	. . . . . 6 |- ( ph -> F/ x ps )
5		nfcv 2178	. . . . . . . 8 |- F/_ x y
6		nfcv 2178	. . . . . . . 8 |- F/_ x z
7	5, 6	nfeq 2185	. . . . . . 7 |- F/ x y = z
8	7	a1i 9	. . . . . 6 |- ( ph -> F/ x y = z )
9	4, 8	nfbid 1480	. . . . 5 |- ( ph -> F/ x ( ps <-> y = z ) )
10	3, 9	nfald 1643	. . . 4 |- ( ph -> F/ x A. y ( ps <-> y = z ) )
11	2, 10	nfabd 2196	. . 3 |- ( ph -> F/_ x { z | A. y ( ps <-> y = z ) } )
12	11	nfunid 3587	. 2 |- ( ph -> F/_ x U. { z | A. y ( ps <-> y = z ) } )
13	1, 12	nfcxfrd 2176	1 |- ( ph -> F/_ x ( iota y ps ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   = wceq 1243   F/wnf 1349   {cab 2026   F/_wnfc 2165   U.cuni 3580   iotacio 4865

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-sn 3381  df-uni 3581  df-iota 4867

This theorem is referenced by:  nfiotaxy  4871  nfriotadxy  5476