Theorem nfiotadw 5254

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 5252	. 2 |- ( iota y ps ) = U. { z | A. y ( ps <-> y = z ) }
2		nfv 1552	. . . 4 |- F/ z ph
3		nfiotadw.1	. . . . 5 |- F/ y ph
4		nfiotadw.2	. . . . . 6 |- ( ph -> F/ x ps )
5		nfcv 2350	. . . . . . . 8 |- F/_ x y
6		nfcv 2350	. . . . . . . 8 |- F/_ x z
7	5, 6	nfeq 2358	. . . . . . 7 |- F/ x y = z
8	7	a1i 9	. . . . . 6 |- ( ph -> F/ x y = z )
9	4, 8	nfbid 1612	. . . . 5 |- ( ph -> F/ x ( ps <-> y = z ) )
10	3, 9	nfald 1784	. . . 4 |- ( ph -> F/ x A. y ( ps <-> y = z ) )
11	2, 10	nfabd 2370	. . 3 |- ( ph -> F/_ x { z | A. y ( ps <-> y = z ) } )
12	11	nfunid 3871	. 2 |- ( ph -> F/_ x U. { z | A. y ( ps <-> y = z ) } )
13	1, 12	nfcxfrd 2348	1 |- ( ph -> F/_ x ( iota y ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   = wceq 1373   F/wnf 1484   {cab 2193   F/_wnfc 2337   U.cuni 3864   iotacio 5249

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-sn 3649  df-uni 3865  df-iota 5251

This theorem is referenced by:  nfiotaw  5255  nfriotadxy  5931