Theorem rmo4f 3004

Description: Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)

Hypotheses

Ref	Expression
rmo4f.1	|- F/_ x A
rmo4f.2	|- F/_ y A
rmo4f.3	|- F/ x ps
rmo4f.4	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
rmo4f	|- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) )

Distinct variable groups:   x, y   ph, y

Allowed substitution hints:   ph( x)   ps( x, y)   A( x, y)

Proof of Theorem rmo4f

Step	Hyp	Ref	Expression
1		rmo4f.1	. . 3 |- F/_ x A
2		rmo4f.2	. . 3 |- F/_ y A
3		nfv 1576	. . 3 |- F/ y ph
4	1, 2, 3	rmo3f 3003	. 2 |- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) )
5		rmo4f.3	. . . . . 6 |- F/ x ps
6		rmo4f.4	. . . . . 6 |- ( x = y -> ( ph <-> ps ) )
7	5, 6	sbie 1839	. . . . 5 |- ( [ y / x ] ph <-> ps )
8	7	anbi2i 457	. . . 4 |- ( ( ph /\ [ y / x ] ph ) <-> ( ph /\ ps ) )
9	8	imbi1i 238	. . 3 |- ( ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> ( ( ph /\ ps ) -> x = y ) )
10	9	2ralbii 2540	. 2 |- ( A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) )
11	4, 10	bitri 184	1 |- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   F/wnf 1508   [wsb 1810   F/_wnfc 2361   A.wral 2510   E*wrmo 2513

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rmo 2518

This theorem is referenced by:  disjxp1  6400