Theorem eumo0 1395

Description: Existential uniqueness implies "at most one."

Hypothesis

Ref	Expression
eumo0.1	|- (ph -> A.yph)

Assertion

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

Distinct variable group:   x,y

Proof of Theorem eumo0

Step	Hyp	Ref	Expression
1		eumo0.1	. . 3 |- (ph -> A.yph)
2	1	euf 1384	. 2 |- (E!xph <-> E.yA.x(ph <-> x = y))
3		bi1 148	. . . 4 |- ((ph <-> x = y) -> (ph -> x = y))
4	3	19.20i 992	. . 3 |- (A.x(ph <-> x = y) -> A.x(ph -> x = y))
5	4	19.22i 1040	. 2 |- (E.yA.x(ph <-> x = y) -> E.yA.x(ph -> x = y))
6	2, 5	sylbi 199	1 |- (E!xph -> E.yA.x(ph -> x = y))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956  E.wex 980  E!weu 1380

This theorem is referenced by:  eu2 1396  mo2 1400

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-eu 1382

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