Theorem reu4 2920

Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.)

Hypothesis

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

Assertion

Ref	Expression
reu4	|- ( E! x e. A ph <-> ( E. x e. A ph /\ A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) ) )

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

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

Proof of Theorem reu4

Step	Hyp	Ref	Expression
1		reu5 2678	. 2 |- ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) )
2		rmo4.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	2	rmo4 2919	. . 3 |- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) )
4	3	anbi2i 453	. 2 |- ( ( E. x e. A ph /\ E* x e. A ph ) <-> ( E. x e. A ph /\ A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) ) )
5	1, 4	bitri 183	1 |- ( E! x e. A ph <-> ( E. x e. A ph /\ A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wral 2444   E.wrex 2445   E!wreu 2446   E*wrmo 2447

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-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-cleq 2158  df-clel 2161  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452

This theorem is referenced by:  reuind  2931  receuap  8566  lbreu  8840  cju  8856  ndvdssub  11867  qredeu  12029