Theorem reusv3i 4437

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 2177	. . . . . 6 |- ( y = z -> ( x = C <-> x = D ) )
4	1, 3	imbi12d 233	. . . . 5 |- ( y = z -> ( ( ph -> x = C ) <-> ( ps -> x = D ) ) )
5	4	cbvralv 2692	. . . 4 |- ( A. y e. B ( ph -> x = C ) <-> A. z e. B ( ps -> x = D ) )
6	5	biimpi 119	. . 3 |- ( A. y e. B ( ph -> x = C ) -> A. z e. B ( ps -> x = D ) )
7		raaanv 3516	. . . 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		anim12 342	. . . . . . 7 |- ( ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> ( ( ph /\ ps ) -> ( x = C /\ x = D ) ) )
9		eqtr2 2184	. . . . . . 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 2529	. . . . 5 |- ( A. z e. B ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> A. z e. B ( ( ph /\ ps ) -> C = D ) )
12	11	ralimi 2529	. . . 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 134	. . 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 418	. 2 |- ( A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
15	14	rexlimivw 2579	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 103   <-> wb 104   = wceq 1343   A.wral 2444   E.wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450

This theorem is referenced by:  reusv3  4438