Theorem nfiotad 4342

Description: Deduction version of nfiota 4343. (Contributed by NM, 18-Feb-2013.)

Hypotheses

Ref	Expression
nfiotad.1	|- F/yph
nfiotad.2	|- (ph -> F/xps)

Assertion

Ref	Expression
nfiotad	|- (ph -> F/_x(iotayps))

Proof of Theorem nfiotad

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiota2 4340	. 2 |- (iotayps) = U.{z | A.y(ps <-> y = z)}
2		nfv 1619	. . . 4 |- F/zph
3		nfiotad.1	. . . . 5 |- F/yph
4		nfiotad.2	. . . . . . 7 |- (ph -> F/xps)
5	4	adantr 451	. . . . . 6 |- ((ph /\ -. A.x x = y) -> F/xps)
6		nfcvf 2511	. . . . . . . 8 |- (-. A.x x = y -> F/_xy)
7	6	adantl 452	. . . . . . 7 |- ((ph /\ -. A.x x = y) -> F/_xy)
8		nfcvd 2490	. . . . . . 7 |- ((ph /\ -. A.x x = y) -> F/_xz)
9	7, 8	nfeqd 2503	. . . . . 6 |- ((ph /\ -. A.x x = y) -> F/x y = z)
10	5, 9	nfbid 1832	. . . . 5 |- ((ph /\ -. A.x x = y) -> F/x(ps <-> y = z))
11	3, 10	nfald2 1972	. . . 4 |- (ph -> F/xA.y(ps <-> y = z))
12	2, 11	nfabd 2508	. . 3 |- (ph -> F/_x{z | A.y(ps <-> y = z)})
13	12	nfunid 3898	. 2 |- (ph -> F/_xU.{z | A.y(ps <-> y = z)})
14	1, 13	nfcxfrd 2487	1 |- (ph -> F/_x(iotayps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   F/wnf 1544   = wceq 1642  {cab 2339   F/_wnfc 2476  U.cuni 3891  iotacio 4337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-sn 3741  df-uni 3892  df-iota 4339

This theorem is referenced by:  nfiota  4343