Theorem euim 1817

Description: Add existential uniqueness quantifiers to an implication. Note the reversed implication in the antecedent. (The proof was 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 4	. . 3 |- (E.xph -> (E!xps -> E.xph))
2		euimmo 1816	. . 3 |- (A.x(ph -> ps) -> (E!xps -> E*xph))
3	1, 2	anim12ii 618	. 2 |- ((E.xph /\ A.x(ph -> ps)) -> (E!xps -> (E.xph /\ E*xph)))
4		eu5 1805	. 2 |- (E!xph <-> (E.xph /\ E*xph))
5	3, 4	syl6ibr 230	1 |- ((E.xph /\ A.x(ph -> ps)) -> (E!xps -> E!xph))

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296  E.wex 1326  E!weu 1771  E*wmo 1772

This theorem is referenced by:  euor2OLD 1840

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776

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