Theorem reu4 2799

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 2574	. 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 2798	. . 3 |- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ ps ) -> x = y ) )
4	3	anbi2i 445	. 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 182	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 102   <-> wb 103   A.wral 2355   E.wrex 2356   E!wreu 2357   E*wrmo 2358

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-cleq 2078  df-clel 2081  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363

This theorem is referenced by:  reuind  2808  receuap  8054  lbreu  8318  cju  8333  ndvdssub  10724  qredeu  10873