Theorem reu7 2998

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 2993	. 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 1758	. . . . . . . 8 |- ( x = y -> ( x = z <-> y = z ) )
4		equcom 1752	. . . . . . . 8 |- ( y = z <-> z = y )
5	3, 4	bitrdi 196	. . . . . . 7 |- ( x = y -> ( x = z <-> z = y ) )
6	2, 5	imbi12d 234	. . . . . 6 |- ( x = y -> ( ( ph -> x = z ) <-> ( ps -> z = y ) ) )
7	6	cbvralv 2765	. . . . 5 |- ( A. x e. A ( ph -> x = z ) <-> A. y e. A ( ps -> z = y ) )
8	7	rexbii 2537	. . . 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 1758	. . . . . . 7 |- ( z = x -> ( z = y <-> x = y ) )
10	9	imbi2d 230	. . . . . 6 |- ( z = x -> ( ( ps -> z = y ) <-> ( ps -> x = y ) ) )
11	10	ralbidv 2530	. . . . 5 |- ( z = x -> ( A. y e. A ( ps -> z = y ) <-> A. y e. A ( ps -> x = y ) ) )
12	11	cbvrexv 2766	. . . 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 184	. . 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 457	. 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 184	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 104   <-> wb 105   A.wral 2508   E.wrex 2509   E!wreu 2510

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516

This theorem is referenced by: (None)