Theorem euex 1387

Description: Existential uniqueness implies existence.

Assertion

Ref	Expression
euex	|- (E!xph -> E.xph)

Proof of Theorem euex

Step	Hyp	Ref	Expression
1		ax-17 968	. . . 4 |- (ph -> A.yph)
2	1	eu1 1385	. . 3 |- (E!xph <-> E.x(ph /\ A.y([y / x]ph -> x = y)))
3		19.40 1090	. . 3 |- (E.x(ph /\ A.y([y / x]ph -> x = y)) -> (E.xph /\ E.xA.y([y / x]ph -> x = y)))
4	2, 3	sylbi 199	. 2 |- (E!xph -> (E.xph /\ E.xA.y([y / x]ph -> x = y)))
5	4	pm3.26d 321	1 |- (E!xph -> E.xph)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951  E.wex 977  E!weu 1373

This theorem is referenced by:  eu2 1389  exmoeu 1406  euor2 1430  2eu2ex 1436  euxfr 1917  reurex 1918  zfrep6 3600  fnopabg 3601  tz6.12c 3725  ndmfv 3730  dff2 3802  fnoprabg 3997  aceq5lem5 4711  hlimeu 9032

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375

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