Theorem euim 2016

Description: Add existential unique existential 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. x ph /\ A. x ( ph -> ps ) ) -> ( E! x ps -> E! x ph ) )

Proof of Theorem euim

Step	Hyp	Ref	Expression
1		ax-1 5	. . 3 |- ( E. x ph -> ( E! x ps -> E. x ph ) )
2		euimmo 2015	. . 3 |- ( A. x ( ph -> ps ) -> ( E! x ps -> E* x ph ) )
3	1, 2	anim12ii 335	. 2 |- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E! x ps -> ( E. x ph /\ E* x ph ) ) )
4		eu5 1995	. 2 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
5	3, 4	syl6ibr 160	1 |- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E! x ps -> E! x ph ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1287   E.wex 1426   E!weu 1948   E*wmo 1949

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952

This theorem is referenced by: (None)