Theorem nfrmo 3115

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

Hypotheses

Ref	Expression
nfreu.1	|- F/_ x A
nfreu.2	|- F/ x ph

Assertion

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

Proof of Theorem nfrmo

Step	Hyp	Ref	Expression
1		df-rmo 2920	. 2 |- ( E* y e. A ph <-> E* y ( y e. A /\ ph ) )
2		nftru 1730	. . . 4 |- F/ y T.
3		nfcvf 2788	. . . . . . 7 |- ( -. A. x x = y -> F/_ x y )
4		nfreu.1	. . . . . . . 8 |- F/_ x A
5	4	a1i 11	. . . . . . 7 |- ( -. A. x x = y -> F/_ x A )
6	3, 5	nfeld 2773	. . . . . 6 |- ( -. A. x x = y -> F/ x y e. A )
7		nfreu.2	. . . . . . 7 |- F/ x ph
8	7	a1i 11	. . . . . 6 |- ( -. A. x x = y -> F/ x ph )
9	6, 8	nfand 1826	. . . . 5 |- ( -. A. x x = y -> F/ x ( y e. A /\ ph ) )
10	9	adantl 482	. . . 4 |- ( ( T. /\ -. A. x x = y ) -> F/ x ( y e. A /\ ph ) )
11	2, 10	nfmod2 2483	. . 3 |- ( T. -> F/ x E* y ( y e. A /\ ph ) )
12	11	trud 1493	. 2 |- F/ x E* y ( y e. A /\ ph )
13	1, 12	nfxfr 1779	1 |- F/ x E* y e. A ph

Syntax hints:   -. wn 3   /\ wa 384   A.wal 1481   T. wtru 1484   F/wnf 1708   e. wcel 1990   E*wmo 2471   F/_wnfc 2751   E*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rmo 2920

This theorem is referenced by:  2rmorex  3412  2reurex  41181