Theorem reu4 2933

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 2690	. 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 2932	. . 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 2455   E.wrex 2456   E!wreu 2457   E*wrmo 2458

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-cleq 2170  df-clel 2173  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463

This theorem is referenced by:  reuind  2944  receuap  8628  lbreu  8904  cju  8920  ndvdssub  11937  qredeu  12099