Theorem rmo4 2445

Description: Restricted "at most one" using implicit substitution.

Hypothesis

Ref	Expression
rmo4.1	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
rmo4	|- (E*x(x e. A /\ ph) <-> A.x e. A A.y e. A ((ph /\ ps) -> x = y))

Distinct variable groups:   x,y,A   ph,y   ps,x

Proof of Theorem rmo4

Step	Hyp	Ref	Expression
1		an4 564	. . . . . . . 8 |- (((x e. A /\ ph) /\ (y e. A /\ ps)) <-> ((x e. A /\ y e. A) /\ (ph /\ ps)))
2		ancom 482	. . . . . . . . 9 |- ((x e. A /\ y e. A) <-> (y e. A /\ x e. A))
3	2	anbi1i 539	. . . . . . . 8 |- (((x e. A /\ y e. A) /\ (ph /\ ps)) <-> ((y e. A /\ x e. A) /\ (ph /\ ps)))
4	1, 3	bitri 190	. . . . . . 7 |- (((x e. A /\ ph) /\ (y e. A /\ ps)) <-> ((y e. A /\ x e. A) /\ (ph /\ ps)))
5	4	imbi1i 203	. . . . . 6 |- ((((x e. A /\ ph) /\ (y e. A /\ ps)) -> x = y) <-> (((y e. A /\ x e. A) /\ (ph /\ ps)) -> x = y))
6		impexp 374	. . . . . 6 |- ((((y e. A /\ x e. A) /\ (ph /\ ps)) -> x = y) <-> ((y e. A /\ x e. A) -> ((ph /\ ps) -> x = y)))
7		impexp 374	. . . . . 6 |- (((y e. A /\ x e. A) -> ((ph /\ ps) -> x = y)) <-> (y e. A -> (x e. A -> ((ph /\ ps) -> x = y))))
8	5, 6, 7	3bitri 194	. . . . 5 |- ((((x e. A /\ ph) /\ (y e. A /\ ps)) -> x = y) <-> (y e. A -> (x e. A -> ((ph /\ ps) -> x = y))))
9	8	albii 1346	. . . 4 |- (A.y(((x e. A /\ ph) /\ (y e. A /\ ps)) -> x = y) <-> A.y(y e. A -> (x e. A -> ((ph /\ ps) -> x = y))))
10		df-ral 2109	. . . 4 |- (A.y e. A (x e. A -> ((ph /\ ps) -> x = y)) <-> A.y(y e. A -> (x e. A -> ((ph /\ ps) -> x = y))))
11		r19.21v 2178	. . . 4 |- (A.y e. A (x e. A -> ((ph /\ ps) -> x = y)) <-> (x e. A -> A.y e. A ((ph /\ ps) -> x = y)))
12	9, 10, 11	3bitr2i 196	. . 3 |- (A.y(((x e. A /\ ph) /\ (y e. A /\ ps)) -> x = y) <-> (x e. A -> A.y e. A ((ph /\ ps) -> x = y)))
13	12	albii 1346	. 2 |- (A.xA.y(((x e. A /\ ph) /\ (y e. A /\ ps)) -> x = y) <-> A.x(x e. A -> A.y e. A ((ph /\ ps) -> x = y)))
14		eleq1 1957	. . . 4 |- (x = y -> (x e. A <-> y e. A))
15		rmo4.1	. . . 4 |- (x = y -> (ph <-> ps))
16	14, 15	anbi12d 690	. . 3 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ ps)))
17	16	mo4 1799	. 2 |- (E*x(x e. A /\ ph) <-> A.xA.y(((x e. A /\ ph) /\ (y e. A /\ ps)) -> x = y))
18		df-ral 2109	. 2 |- (A.x e. A A.y e. A ((ph /\ ps) -> x = y) <-> A.x(x e. A -> A.y e. A ((ph /\ ps) -> x = y)))
19	13, 17, 18	3bitr4i 200	1 |- (E*x(x e. A /\ ph) <-> A.x e. A A.y e. A ((ph /\ ps) -> x = y))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E*wmo 1772  A.wral 2105

This theorem is referenced by:  reu4 2446

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-cleq 1877  df-clel 1880  df-ral 2109

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