Theorem reusv3i 4209

Description: Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)

Hypotheses

Ref	Expression
reusv3.1	|- ( y = z -> ( ph <-> ps ) )
reusv3.2	|- ( y = z -> C = D )

Assertion

Ref	Expression
reusv3i	|- ( E. x e. A A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )

Distinct variable groups:   x, y, z, B   x, C, z   x, D, y   ph, x, z   ps, x, y

Allowed substitution hints:   ph( y)   ps( z)   A( x, y, z)   C( y)   D( z)

Proof of Theorem reusv3i

Step	Hyp	Ref	Expression
1		reusv3.1	. . . . . 6 |- ( y = z -> ( ph <-> ps ) )
2		reusv3.2	. . . . . . 7 |- ( y = z -> C = D )
3	2	eqeq2d 2092	. . . . . 6 |- ( y = z -> ( x = C <-> x = D ) )
4	1, 3	imbi12d 232	. . . . 5 |- ( y = z -> ( ( ph -> x = C ) <-> ( ps -> x = D ) ) )
5	4	cbvralv 2577	. . . 4 |- ( A. y e. B ( ph -> x = C ) <-> A. z e. B ( ps -> x = D ) )
6	5	biimpi 118	. . 3 |- ( A. y e. B ( ph -> x = C ) -> A. z e. B ( ps -> x = D ) )
7		raaanv 3348	. . . 4 |- ( A. y e. B A. z e. B ( ( ph -> x = C ) /\ ( ps -> x = D ) ) <-> ( A. y e. B ( ph -> x = C ) /\ A. z e. B ( ps -> x = D ) ) )
8		prth 336	. . . . . . 7 |- ( ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> ( ( ph /\ ps ) -> ( x = C /\ x = D ) ) )
9		eqtr2 2099	. . . . . . 7 |- ( ( x = C /\ x = D ) -> C = D )
10	8, 9	syl6 33	. . . . . 6 |- ( ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> ( ( ph /\ ps ) -> C = D ) )
11	10	ralimi 2426	. . . . 5 |- ( A. z e. B ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> A. z e. B ( ( ph /\ ps ) -> C = D ) )
12	11	ralimi 2426	. . . 4 |- ( A. y e. B A. z e. B ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
13	7, 12	sylbir 133	. . 3 |- ( ( A. y e. B ( ph -> x = C ) /\ A. z e. B ( ps -> x = D ) ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
14	6, 13	mpdan 412	. 2 |- ( A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
15	14	rexlimivw 2473	1 |- ( E. x e. A A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   A.wral 2348   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354

This theorem is referenced by:  reusv3  4210