Theorem euim 2082

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 6	. . 3 |- ( E. x ph -> ( E! x ps -> E. x ph ) )
2		euimmo 2081	. . 3 |- ( A. x ( ph -> ps ) -> ( E! x ps -> E* x ph ) )
3	1, 2	anim12ii 341	. 2 |- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E! x ps -> ( E. x ph /\ E* x ph ) ) )
4		eu5 2061	. 2 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
5	3, 4	syl6ibr 161	1 |- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E! x ps -> E! x ph ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1341   E.wex 1480   E!weu 2014   E*wmo 2015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018

This theorem is referenced by: (None)