Theorem euim 2254

Description: Add existential uniqueness quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
euim	|- ((E.xph /\ A.x(ph -> ps)) -> (E!xps -> E!xph))

Proof of Theorem euim

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 |- (E.xph -> (E!xps -> E.xph))
2		euimmo 2253	. . 3 |- (A.x(ph -> ps) -> (E!xps -> E*xph))
3	1, 2	anim12ii 553	. 2 |- ((E.xph /\ A.x(ph -> ps)) -> (E!xps -> (E.xph /\ E*xph)))
4		eu5 2242	. 2 |- (E!xph <-> (E.xph /\ E*xph))
5	3, 4	syl6ibr 218	1 |- ((E.xph /\ A.x(ph -> ps)) -> (E!xps -> E!xph))

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541  E!weu 2204  E*wmo 2205

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

This theorem is referenced by: (None)