Theorem rmo4 3030

Description: Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.)

Hypothesis

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

Assertion

Ref	Expression
rmo4	|- (E*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

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

Proof of Theorem rmo4

Step	Hyp	Ref	Expression
1		df-rmo 2623	. 2 |- (E*x e. A ph <-> E*x(x e. A /\ ph))
2		an4 797	. . . . . . . . 9 |- (((x e. A /\ ph) /\ (y e. A /\ ps)) <-> ((x e. A /\ y e. A) /\ (ph /\ ps)))
3		ancom 437	. . . . . . . . . 10 |- ((x e. A /\ y e. A) <-> (y e. A /\ x e. A))
4	3	anbi1i 676	. . . . . . . . 9 |- (((x e. A /\ y e. A) /\ (ph /\ ps)) <-> ((y e. A /\ x e. A) /\ (ph /\ ps)))
5	2, 4	bitri 240	. . . . . . . 8 |- (((x e. A /\ ph) /\ (y e. A /\ ps)) <-> ((y e. A /\ x e. A) /\ (ph /\ ps)))
6	5	imbi1i 315	. . . . . . 7 |- ((((x e. A /\ ph) /\ (y e. A /\ ps)) -> x = y) <-> (((y e. A /\ x e. A) /\ (ph /\ ps)) -> x = y))
7		impexp 433	. . . . . . 7 |- ((((y e. A /\ x e. A) /\ (ph /\ ps)) -> x = y) <-> ((y e. A /\ x e. A) -> ((ph /\ ps) -> x = y)))
8		impexp 433	. . . . . . 7 |- (((y e. A /\ x e. A) -> ((ph /\ ps) -> x = y)) <-> (y e. A -> (x e. A -> ((ph /\ ps) -> x = y))))
9	6, 7, 8	3bitri 262	. . . . . 6 |- ((((x e. A /\ ph) /\ (y e. A /\ ps)) -> x = y) <-> (y e. A -> (x e. A -> ((ph /\ ps) -> x = y))))
10	9	albii 1566	. . . . 5 |- (A.y(((x e. A /\ ph) /\ (y e. A /\ ps)) -> x = y) <-> A.y(y e. A -> (x e. A -> ((ph /\ ps) -> x = y))))
11		df-ral 2620	. . . . 5 |- (A.y e. A (x e. A -> ((ph /\ ps) -> x = y)) <-> A.y(y e. A -> (x e. A -> ((ph /\ ps) -> x = y))))
12		r19.21v 2702	. . . . 5 |- (A.y e. A (x e. A -> ((ph /\ ps) -> x = y)) <-> (x e. A -> A.y e. A ((ph /\ ps) -> x = y)))
13	10, 11, 12	3bitr2i 264	. . . 4 |- (A.y(((x e. A /\ ph) /\ (y e. A /\ ps)) -> x = y) <-> (x e. A -> A.y e. A ((ph /\ ps) -> x = y)))
14	13	albii 1566	. . 3 |- (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)))
15		eleq1 2413	. . . . 5 |- (x = y -> (x e. A <-> y e. A))
16		rmo4.1	. . . . 5 |- (x = y -> (ph <-> ps))
17	15, 16	anbi12d 691	. . . 4 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ ps)))
18	17	mo4 2237	. . 3 |- (E*x(x e. A /\ ph) <-> A.xA.y(((x e. A /\ ph) /\ (y e. A /\ ps)) -> x = y))
19		df-ral 2620	. . 3 |- (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)))
20	14, 18, 19	3bitr4i 268	. 2 |- (E*x(x e. A /\ ph) <-> A.x e. A A.y e. A ((ph /\ ps) -> x = y))
21	1, 20	bitri 240	1 |- (E*x e. A ph <-> A.x e. A A.y e. A ((ph /\ ps) -> x = y))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   e. wcel 1710  E*wmo 2205  A.wral 2615  E*wrmo 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-cleq 2346  df-clel 2349  df-ral 2620  df-rmo 2623

This theorem is referenced by:  reu4  3031