Theorem reusv3i 4217

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 2093	. . . . . 6 |- ( y = z -> ( x = C <-> x = D ) )
4	1, 3	imbi12d 232	. . . . 5 |- ( y = z -> ( ( ph -> x = C ) <-> ( ps -> x = D ) ) )
5	4	cbvralv 2578	. . . 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 3356	. . . 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 2100	. . . . . . 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 2427	. . . . 5 |- ( A. z e. B ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> A. z e. B ( ( ph /\ ps ) -> C = D ) )
12	11	ralimi 2427	. . . 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 2474	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 1285   A.wral 2349   E.wrex 2350

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355

This theorem is referenced by:  reusv3  4218