Theorem nfiotadw 5173

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 5171	. 2 |- ( iota y ps ) = U. { z | A. y ( ps <-> y = z ) }
2		nfv 1526	. . . 4 |- F/ z ph
3		nfiotadw.1	. . . . 5 |- F/ y ph
4		nfiotadw.2	. . . . . 6 |- ( ph -> F/ x ps )
5		nfcv 2317	. . . . . . . 8 |- F/_ x y
6		nfcv 2317	. . . . . . . 8 |- F/_ x z
7	5, 6	nfeq 2325	. . . . . . 7 |- F/ x y = z
8	7	a1i 9	. . . . . 6 |- ( ph -> F/ x y = z )
9	4, 8	nfbid 1586	. . . . 5 |- ( ph -> F/ x ( ps <-> y = z ) )
10	3, 9	nfald 1758	. . . 4 |- ( ph -> F/ x A. y ( ps <-> y = z ) )
11	2, 10	nfabd 2337	. . 3 |- ( ph -> F/_ x { z | A. y ( ps <-> y = z ) } )
12	11	nfunid 3812	. 2 |- ( ph -> F/_ x U. { z | A. y ( ps <-> y = z ) } )
13	1, 12	nfcxfrd 2315	1 |- ( ph -> F/_ x ( iota y ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351   = wceq 1353   F/wnf 1458   {cab 2161   F/_wnfc 2304   U.cuni 3805   iotacio 5168

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-sn 3595  df-uni 3806  df-iota 5170

This theorem is referenced by:  nfiotaw  5174  nfriotadxy  5829