Theorem reu4 1937

Description: Restricted uniqueness using implicit substitution.

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

Proof of Theorem reu4

Step	Hyp	Ref	Expression
1		reu5 1932	. 2 |- (E!x e. A ph <-> (E.x e. A ph /\ E*x(x e. A /\ ph)))
2		rmo4.1	. . . 4 |- (x = y -> (ph <-> ps))
3	2	rmo4 1936	. . 3 |- (E*x(x e. A /\ ph) <-> A.x e. A A.y e. A ((ph /\ ps) -> x = y))
4	3	anbi2i 482	. 2 |- ((E.x e. A ph /\ E*x(x e. A /\ ph)) <-> (E.x e. A ph /\ A.x e. A A.y e. A ((ph /\ ps) -> x = y)))
5	1, 4	bitr 173	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 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E*wmo 1383  A.wral 1648  E.wrex 1649  E!wreu 1650

This theorem is referenced by:  wereu 2951  oawordeulem 4194  negeu 5367  receu 5713  lbreu 6047  uzwo5OLD 6213  creur 6743  creui 6744  minveceu 8579  hlimreu 9105  pjtheu 9230  adjvalvalt 9856  riesz4 9991  cdjreu 10354

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-reu 1654

Copyright terms: Public domain