Theorem rmo4 1936

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 508	. . . . . . . 8 |- (((x e. A /\ ph) /\ (y e. A /\ ps)) <-> ((x e. A /\ y e. A) /\ (ph /\ ps)))
2		ancom 437	. . . . . . . . 9 |- ((x e. A /\ y e. A) <-> (y e. A /\ x e. A))
3	2	anbi1i 483	. . . . . . . 8 |- (((x e. A /\ y e. A) /\ (ph /\ ps)) <-> ((y e. A /\ x e. A) /\ (ph /\ ps)))
4	1, 3	bitr 173	. . . . . . 7 |- (((x e. A /\ ph) /\ (y e. A /\ ps)) <-> ((y e. A /\ x e. A) /\ (ph /\ ps)))
5	4	imbi1i 186	. . . . . 6 |- ((((x e. A /\ ph) /\ (y e. A /\ ps)) -> x = y) <-> (((y e. A /\ x e. A) /\ (ph /\ ps)) -> x = y))
6		impexp 347	. . . . . 6 |- ((((y e. A /\ x e. A) /\ (ph /\ ps)) -> x = y) <-> ((y e. A /\ x e. A) -> ((ph /\ ps) -> x = y)))
7		impexp 347	. . . . . 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	3bitr 177	. . . . 5 |- ((((x e. A /\ ph) /\ (y e. A /\ ps)) -> x = y) <-> (y e. A -> (x e. A -> ((ph /\ ps) -> x = y))))
9	8	albii 1001	. . . 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 1652	. . . 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 1719	. . . 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	3bitr2 179	. . 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 1001	. 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 1537	. . . 4 |- (x = y -> (x e. A <-> y e. A))
15		rmo4.1	. . . 4 |- (x = y -> (ph <-> ps))
16	14, 15	anbi12d 630	. . 3 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ ps)))
17	16	mo4 1405	. 2 |- (E*x(x e. A /\ ph) <-> A.xA.y(((x e. A /\ ph) /\ (y e. A /\ ps)) -> x = y))
18		df-ral 1652	. 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	3bitr4 183	1 |- (E*x(x e. A /\ ph) <-> A.x e. A A.y e. A ((ph /\ ps) -> x = y))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E*wmo 1383  A.wral 1648

This theorem is referenced by:  reu4 1937

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-cleq 1472  df-clel 1475  df-ral 1652

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