Theorem eueq2dc 2999

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

Hypotheses

Ref	Expression
eueq2dc.1	|- A e. _V
eueq2dc.2	|- B e. _V

Assertion

Ref	Expression
eueq2dc	|- (DECID ph -> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) )

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

Proof of Theorem eueq2dc

Step	Hyp	Ref	Expression
1		df-dc 847	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		notnot 638	. . . . 5 |- ( ph -> -. -. ph )
3		eueq2dc.1	. . . . . . 7 |- A e. _V
4	3	eueq1 2998	. . . . . 6 |- E! x x = A
5		euanv 2144	. . . . . . 7 |- ( E! x ( ph /\ x = A ) <-> ( ph /\ E! x x = A ) )
6	5	biimpri 133	. . . . . 6 |- ( ( ph /\ E! x x = A ) -> E! x ( ph /\ x = A ) )
7	4, 6	mpan2 429	. . . . 5 |- ( ph -> E! x ( ph /\ x = A ) )
8		euorv 2113	. . . . 5 |- ( ( -. -. ph /\ E! x ( ph /\ x = A ) ) -> E! x ( -. ph \/ ( ph /\ x = A ) ) )
9	2, 7, 8	syl2anc 415	. . . 4 |- ( ph -> E! x ( -. ph \/ ( ph /\ x = A ) ) )
10		orcom 740	. . . . . 6 |- ( ( -. ph \/ ( ph /\ x = A ) ) <-> ( ( ph /\ x = A ) \/ -. ph ) )
11	2	bianfd 961	. . . . . . 7 |- ( ph -> ( -. ph <-> ( -. ph /\ x = B ) ) )
12	11	orbi2d 802	. . . . . 6 |- ( ph -> ( ( ( ph /\ x = A ) \/ -. ph ) <-> ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
13	10, 12	bitrid 192	. . . . 5 |- ( ph -> ( ( -. ph \/ ( ph /\ x = A ) ) <-> ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
14	13	eubidv 2094	. . . 4 |- ( ph -> ( E! x ( -. ph \/ ( ph /\ x = A ) ) <-> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
15	9, 14	mpbid 147	. . 3 |- ( ph -> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) )
16		eueq2dc.2	. . . . . . 7 |- B e. _V
17	16	eueq1 2998	. . . . . 6 |- E! x x = B
18		euanv 2144	. . . . . . 7 |- ( E! x ( -. ph /\ x = B ) <-> ( -. ph /\ E! x x = B ) )
19	18	biimpri 133	. . . . . 6 |- ( ( -. ph /\ E! x x = B ) -> E! x ( -. ph /\ x = B ) )
20	17, 19	mpan2 429	. . . . 5 |- ( -. ph -> E! x ( -. ph /\ x = B ) )
21		euorv 2113	. . . . 5 |- ( ( -. ph /\ E! x ( -. ph /\ x = B ) ) -> E! x ( ph \/ ( -. ph /\ x = B ) ) )
22	20, 21	mpdan 425	. . . 4 |- ( -. ph -> E! x ( ph \/ ( -. ph /\ x = B ) ) )
23		id 19	. . . . . . 7 |- ( -. ph -> -. ph )
24	23	bianfd 961	. . . . . 6 |- ( -. ph -> ( ph <-> ( ph /\ x = A ) ) )
25	24	orbi1d 803	. . . . 5 |- ( -. ph -> ( ( ph \/ ( -. ph /\ x = B ) ) <-> ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
26	25	eubidv 2094	. . . 4 |- ( -. ph -> ( E! x ( ph \/ ( -. ph /\ x = B ) ) <-> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
27	22, 26	mpbid 147	. . 3 |- ( -. ph -> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) )
28	15, 27	jaoi 728	. 2 |- ( ( ph \/ -. ph ) -> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) )
29	1, 28	sylbi 121	1 |- (DECID ph -> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   = wceq 1402   E!weu 2086   e. wcel 2209   _Vcvv 2821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823

This theorem is referenced by: (None)