Theorem rmo4f 29337

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 1843	. . 3 |- F/ y ph
4	1, 2, 3	rmo3f 29335	. 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 2408	. . . . 5 |- ( [ y / x ] ph <-> ps )
8	7	anbi2i 730	. . . 4 |- ( ( ph /\ [ y / x ] ph ) <-> ( ph /\ ps ) )
9	8	imbi1i 339	. . 3 |- ( ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> ( ( ph /\ ps ) -> x = y ) )
10	9	2ralbii 2981	. 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 264	1 |- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   F/wnf 1708   [wsb 1880   F/_wnfc 2751   A.wral 2912   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-sb 1881  df-eu 2474  df-mo 2475  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rmo 2920

This theorem is referenced by:  disjorf  29392  funcnv5mpt  29469