Theorem reu4 2974

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 2726	. 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 2973	. . 3 |- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) )
4	3	anbi2i 457	. 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 184	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 104   <-> wb 105   A.wral 2486   E.wrex 2487   E!wreu 2488   E*wrmo 2489

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-cleq 2200  df-clel 2203  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494

This theorem is referenced by:  reuind  2985  receuap  8777  lbreu  9053  cju  9069  ndvdssub  12356  qredeu  12534