Theorem nfrmo 2786

Description: Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 16-Jun-2017.)

Hypotheses

Ref	Expression
nfreu.1	|- F/_xA
nfreu.2	|- F/xph

Assertion

Ref	Expression
nfrmo	|- F/xE*y e. A ph

Proof of Theorem nfrmo

Step	Hyp	Ref	Expression
1		df-rmo 2622	. 2 |- (E*y e. A ph <-> E*y(y e. A /\ ph))
2		nftru 1554	. . . 4 |- F/y T.
3		nfcvf 2511	. . . . . . 7 |- (-. A.x x = y -> F/_xy)
4		nfreu.1	. . . . . . . 8 |- F/_xA
5	4	a1i 10	. . . . . . 7 |- (-. A.x x = y -> F/_xA)
6	3, 5	nfeld 2504	. . . . . 6 |- (-. A.x x = y -> F/x y e. A)
7		nfreu.2	. . . . . . 7 |- F/xph
8	7	a1i 10	. . . . . 6 |- (-. A.x x = y -> F/xph)
9	6, 8	nfand 1822	. . . . 5 |- (-. A.x x = y -> F/x(y e. A /\ ph))
10	9	adantl 452	. . . 4 |- (( T. /\ -. A.x x = y) -> F/x(y e. A /\ ph))
11	2, 10	nfmod2 2217	. . 3 |- ( T. -> F/xE*y(y e. A /\ ph))
12	11	trud 1323	. 2 |- F/xE*y(y e. A /\ ph)
13	1, 12	nfxfr 1570	1 |- F/xE*y e. A ph

Syntax hints:  -. wn 3   /\ wa 358   T. wtru 1316  A.wal 1540   F/wnf 1544   e. wcel 1710  E*wmo 2205   F/_wnfc 2476  E*wrmo 2617

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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rmo 2622

This theorem is referenced by:  2rmorex  3040