Theorem eueq2 3380

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		notnot 136	. . . 4 |- ( ph -> -. -. ph )
2		eueq2.1	. . . . . 6 |- A e. _V
3	2	eueq1 3379	. . . . 5 |- E! x x = A
4		euanv 2534	. . . . . 6 |- ( E! x ( ph /\ x = A ) <-> ( ph /\ E! x x = A ) )
5	4	biimpri 218	. . . . 5 |- ( ( ph /\ E! x x = A ) -> E! x ( ph /\ x = A ) )
6	3, 5	mpan2 707	. . . 4 |- ( ph -> E! x ( ph /\ x = A ) )
7		euorv 2513	. . . 4 |- ( ( -. -. ph /\ E! x ( ph /\ x = A ) ) -> E! x ( -. ph \/ ( ph /\ x = A ) ) )
8	1, 6, 7	syl2anc 693	. . 3 |- ( ph -> E! x ( -. ph \/ ( ph /\ x = A ) ) )
9		orcom 402	. . . . 5 |- ( ( -. ph \/ ( ph /\ x = A ) ) <-> ( ( ph /\ x = A ) \/ -. ph ) )
10	1	bianfd 967	. . . . . 6 |- ( ph -> ( -. ph <-> ( -. ph /\ x = B ) ) )
11	10	orbi2d 738	. . . . 5 |- ( ph -> ( ( ( ph /\ x = A ) \/ -. ph ) <-> ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
12	9, 11	syl5bb 272	. . . 4 |- ( ph -> ( ( -. ph \/ ( ph /\ x = A ) ) <-> ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
13	12	eubidv 2490	. . 3 |- ( ph -> ( E! x ( -. ph \/ ( ph /\ x = A ) ) <-> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
14	8, 13	mpbid 222	. 2 |- ( ph -> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) )
15		eueq2.2	. . . . . 6 |- B e. _V
16	15	eueq1 3379	. . . . 5 |- E! x x = B
17		euanv 2534	. . . . . 6 |- ( E! x ( -. ph /\ x = B ) <-> ( -. ph /\ E! x x = B ) )
18	17	biimpri 218	. . . . 5 |- ( ( -. ph /\ E! x x = B ) -> E! x ( -. ph /\ x = B ) )
19	16, 18	mpan2 707	. . . 4 |- ( -. ph -> E! x ( -. ph /\ x = B ) )
20		euorv 2513	. . . 4 |- ( ( -. ph /\ E! x ( -. ph /\ x = B ) ) -> E! x ( ph \/ ( -. ph /\ x = B ) ) )
21	19, 20	mpdan 702	. . 3 |- ( -. ph -> E! x ( ph \/ ( -. ph /\ x = B ) ) )
22		id 22	. . . . . 6 |- ( -. ph -> -. ph )
23	22	bianfd 967	. . . . 5 |- ( -. ph -> ( ph <-> ( ph /\ x = A ) ) )
24	23	orbi1d 739	. . . 4 |- ( -. ph -> ( ( ph \/ ( -. ph /\ x = B ) ) <-> ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
25	24	eubidv 2490	. . 3 |- ( -. ph -> ( E! x ( ph \/ ( -. ph /\ x = B ) ) <-> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
26	21, 25	mpbid 222	. 2 |- ( -. ph -> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) )
27	14, 26	pm2.61i 176	1 |- E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) )

Syntax hints:   -. wn 3   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   E!weu 2470   _Vcvv 3200

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-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by: (None)