Theorem eueq2 3011

Description: Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)

Hypotheses

Ref	Expression
eueq2.1	|- A e. _V
eueq2.2	|- B e. _V

Assertion

Ref	Expression
eueq2	|- E!x((ph /\ x = A) \/ (-. ph /\ x = B))

Distinct variable groups:   ph,x   x,A   x,B

Proof of Theorem eueq2

Step	Hyp	Ref	Expression
1		notnot1 114	. . . 4 |- (ph -> -. -. ph)
2		eueq2.1	. . . . . 6 |- A e. _V
3	2	eueq1 3010	. . . . 5 |- E!x x = A
4		euanv 2265	. . . . . 6 |- (E!x(ph /\ x = A) <-> (ph /\ E!x x = A))
5	4	biimpri 197	. . . . 5 |- ((ph /\ E!x x = A) -> E!x(ph /\ x = A))
6	3, 5	mpan2 652	. . . 4 |- (ph -> E!x(ph /\ x = A))
7		euorv 2232	. . . 4 |- ((-. -. ph /\ E!x(ph /\ x = A)) -> E!x(-. ph \/ (ph /\ x = A)))
8	1, 6, 7	syl2anc 642	. . 3 |- (ph -> E!x(-. ph \/ (ph /\ x = A)))
9		orcom 376	. . . . 5 |- ((-. ph \/ (ph /\ x = A)) <-> ((ph /\ x = A) \/ -. ph))
10	1	bianfd 892	. . . . . 6 |- (ph -> (-. ph <-> (-. ph /\ x = B)))
11	10	orbi2d 682	. . . . 5 |- (ph -> (((ph /\ x = A) \/ -. ph) <-> ((ph /\ x = A) \/ (-. ph /\ x = B))))
12	9, 11	syl5bb 248	. . . 4 |- (ph -> ((-. ph \/ (ph /\ x = A)) <-> ((ph /\ x = A) \/ (-. ph /\ x = B))))
13	12	eubidv 2212	. . 3 |- (ph -> (E!x(-. ph \/ (ph /\ x = A)) <-> E!x((ph /\ x = A) \/ (-. ph /\ x = B))))
14	8, 13	mpbid 201	. 2 |- (ph -> E!x((ph /\ x = A) \/ (-. ph /\ x = B)))
15		eueq2.2	. . . . . 6 |- B e. _V
16	15	eueq1 3010	. . . . 5 |- E!x x = B
17		euanv 2265	. . . . . 6 |- (E!x(-. ph /\ x = B) <-> (-. ph /\ E!x x = B))
18	17	biimpri 197	. . . . 5 |- ((-. ph /\ E!x x = B) -> E!x(-. ph /\ x = B))
19	16, 18	mpan2 652	. . . 4 |- (-. ph -> E!x(-. ph /\ x = B))
20		euorv 2232	. . . 4 |- ((-. ph /\ E!x(-. ph /\ x = B)) -> E!x(ph \/ (-. ph /\ x = B)))
21	19, 20	mpdan 649	. . 3 |- (-. ph -> E!x(ph \/ (-. ph /\ x = B)))
22		id 19	. . . . . 6 |- (-. ph -> -. ph)
23	22	bianfd 892	. . . . 5 |- (-. ph -> (ph <-> (ph /\ x = A)))
24	23	orbi1d 683	. . . 4 |- (-. ph -> ((ph \/ (-. ph /\ x = B)) <-> ((ph /\ x = A) \/ (-. ph /\ x = B))))
25	24	eubidv 2212	. . 3 |- (-. ph -> (E!x(ph \/ (-. ph /\ x = B)) <-> E!x((ph /\ x = A) \/ (-. ph /\ x = B))))
26	21, 25	mpbid 201	. 2 |- (-. ph -> E!x((ph /\ x = A) \/ (-. ph /\ x = B)))
27	14, 26	pm2.61i 156	1 |- E!x((ph /\ x = A) \/ (-. ph /\ x = B))

Syntax hints:  -. wn 3   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710  E!weu 2204  _Vcvv 2860

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-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by: (None)