Theorem reu7 2921

Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)

Hypothesis

Ref	Expression
rmo4.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
reu7	|- ( E! x e. A ph <-> ( E. x e. A ph /\ E. x e. A A. y e. A ( ps -> x = y ) ) )

Distinct variable groups:   x, y, A   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem reu7

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reu3 2916	. 2 |- ( E! x e. A ph <-> ( E. x e. A ph /\ E. z e. A A. x e. A ( ph -> x = z ) ) )
2		rmo4.1	. . . . . . 7 |- ( x = y -> ( ph <-> ps ) )
3		equequ1 1700	. . . . . . . 8 |- ( x = y -> ( x = z <-> y = z ) )
4		equcom 1694	. . . . . . . 8 |- ( y = z <-> z = y )
5	3, 4	bitrdi 195	. . . . . . 7 |- ( x = y -> ( x = z <-> z = y ) )
6	2, 5	imbi12d 233	. . . . . 6 |- ( x = y -> ( ( ph -> x = z ) <-> ( ps -> z = y ) ) )
7	6	cbvralv 2692	. . . . 5 |- ( A. x e. A ( ph -> x = z ) <-> A. y e. A ( ps -> z = y ) )
8	7	rexbii 2473	. . . 4 |- ( E. z e. A A. x e. A ( ph -> x = z ) <-> E. z e. A A. y e. A ( ps -> z = y ) )
9		equequ1 1700	. . . . . . 7 |- ( z = x -> ( z = y <-> x = y ) )
10	9	imbi2d 229	. . . . . 6 |- ( z = x -> ( ( ps -> z = y ) <-> ( ps -> x = y ) ) )
11	10	ralbidv 2466	. . . . 5 |- ( z = x -> ( A. y e. A ( ps -> z = y ) <-> A. y e. A ( ps -> x = y ) ) )
12	11	cbvrexv 2693	. . . 4 |- ( E. z e. A A. y e. A ( ps -> z = y ) <-> E. x e. A A. y e. A ( ps -> x = y ) )
13	8, 12	bitri 183	. . 3 |- ( E. z e. A A. x e. A ( ph -> x = z ) <-> E. x e. A A. y e. A ( ps -> x = y ) )
14	13	anbi2i 453	. 2 |- ( ( E. x e. A ph /\ E. z e. A A. x e. A ( ph -> x = z ) ) <-> ( E. x e. A ph /\ E. x e. A A. y e. A ( ps -> x = y ) ) )
15	1, 14	bitri 183	1 |- ( E! x e. A ph <-> ( E. x e. A ph /\ E. x e. A A. y e. A ( ps -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wral 2444   E.wrex 2445   E!wreu 2446

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-eu 2017  df-mo 2018  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452

This theorem is referenced by: (None)