Theorem nfiotadw 5156

Description: Bound-variable hypothesis builder for the iota class. (Contributed by Jim Kingdon, 21-Dec-2018.)

Hypotheses

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

Assertion

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

Distinct variable group:   x, y

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

Proof of Theorem nfiotadw

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiota2 5154	. 2 |- ( iota y ps ) = U. { z | A. y ( ps <-> y = z ) }
2		nfv 1516	. . . 4 |- F/ z ph
3		nfiotadw.1	. . . . 5 |- F/ y ph
4		nfiotadw.2	. . . . . 6 |- ( ph -> F/ x ps )
5		nfcv 2308	. . . . . . . 8 |- F/_ x y
6		nfcv 2308	. . . . . . . 8 |- F/_ x z
7	5, 6	nfeq 2316	. . . . . . 7 |- F/ x y = z
8	7	a1i 9	. . . . . 6 |- ( ph -> F/ x y = z )
9	4, 8	nfbid 1576	. . . . 5 |- ( ph -> F/ x ( ps <-> y = z ) )
10	3, 9	nfald 1748	. . . 4 |- ( ph -> F/ x A. y ( ps <-> y = z ) )
11	2, 10	nfabd 2328	. . 3 |- ( ph -> F/_ x { z | A. y ( ps <-> y = z ) } )
12	11	nfunid 3796	. 2 |- ( ph -> F/_ x U. { z | A. y ( ps <-> y = z ) } )
13	1, 12	nfcxfrd 2306	1 |- ( ph -> F/_ x ( iota y ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   F/wnf 1448   {cab 2151   F/_wnfc 2295   U.cuni 3789   iotacio 5151

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-sn 3582  df-uni 3790  df-iota 5153

This theorem is referenced by:  nfiotaw  5157  nfriotadxy  5806