Theorem eumo0 2228

Description: Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)

Hypothesis

Ref	Expression
eumo0.1	|- F/yph

Assertion

Ref	Expression
eumo0	|- (E!xph -> E.yA.x(ph -> x = y))

Distinct variable group:   x,y

Allowed substitution hints:   ph(x,y)

Proof of Theorem eumo0

Step	Hyp	Ref	Expression
1		eumo0.1	. . 3 |- F/yph
2	1	euf 2210	. 2 |- (E!xph <-> E.yA.x(ph <-> x = y))
3		bi1 178	. . . 4 |- ((ph <-> x = y) -> (ph -> x = y))
4	3	alimi 1559	. . 3 |- (A.x(ph <-> x = y) -> A.x(ph -> x = y))
5	4	eximi 1576	. 2 |- (E.yA.x(ph <-> x = y) -> E.yA.x(ph -> x = y))
6	2, 5	sylbi 187	1 |- (E!xph -> E.yA.x(ph -> x = y))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541   F/wnf 1544  E!weu 2204

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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208

This theorem is referenced by:  eu2  2229  mo2  2233