Theorem rmo2 3132

Description: Alternate definition of restricted "at most one." Note that E*x e. Aph is not equivalent to E.y e. AA.x e. A(ph -> x = y) (in analogy to reu6 3026); to see this, let A be the empty set. However, one direction of this pattern holds; see rmo2i 3133. (Contributed by NM, 17-Jun-2017.)

Hypothesis

Ref	Expression
rmo2.1	|- F/yph

Assertion

Ref	Expression
rmo2	|- (E*x e. A ph <-> E.yA.x e. A (ph -> x = y))

Distinct variable group:   x,y,A

Allowed substitution hints:   ph(x,y)

Proof of Theorem rmo2

Step	Hyp	Ref	Expression
1		df-rmo 2623	. 2 |- (E*x e. A ph <-> E*x(x e. A /\ ph))
2		nfv 1619	. . . 4 |- F/y x e. A
3		rmo2.1	. . . 4 |- F/yph
4	2, 3	nfan 1824	. . 3 |- F/y(x e. A /\ ph)
5	4	mo2 2233	. 2 |- (E*x(x e. A /\ ph) <-> E.yA.x((x e. A /\ ph) -> x = y))
6		impexp 433	. . . . 5 |- (((x e. A /\ ph) -> x = y) <-> (x e. A -> (ph -> x = y)))
7	6	albii 1566	. . . 4 |- (A.x((x e. A /\ ph) -> x = y) <-> A.x(x e. A -> (ph -> x = y)))
8		df-ral 2620	. . . 4 |- (A.x e. A (ph -> x = y) <-> A.x(x e. A -> (ph -> x = y)))
9	7, 8	bitr4i 243	. . 3 |- (A.x((x e. A /\ ph) -> x = y) <-> A.x e. A (ph -> x = y))
10	9	exbii 1582	. 2 |- (E.yA.x((x e. A /\ ph) -> x = y) <-> E.yA.x e. A (ph -> x = y))
11	1, 5, 10	3bitri 262	1 |- (E*x e. A ph <-> E.yA.x e. A (ph -> x = y))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   F/wnf 1544   e. wcel 1710  E*wmo 2205  A.wral 2615  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

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-ral 2620  df-rmo 2623

This theorem is referenced by:  rmo2i  3133