Theorem reu4 3031

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 2825	. 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 3030	. . 3 |- (E*x e. A ph <-> A.x e. A A.y e. A ((ph /\ ps) -> x = y))
4	3	anbi2i 675	. 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 240	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   <-> wb 176   /\ wa 358  A.wral 2615  E.wrex 2616  E!wreu 2617  E*wrmo 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-cleq 2346  df-clel 2349  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623

This theorem is referenced by:  reuind  3040  vfinnc  4472  nnpw1ex  4485  ncspw1eu  6160