Theorem euim 2122

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 2121	. . 3 |- ( A. x ( ph -> ps ) -> ( E! x ps -> E* x ph ) )
3	1, 2	anim12ii 343	. 2 |- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E! x ps -> ( E. x ph /\ E* x ph ) ) )
4		eu5 2101	. 2 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
5	3, 4	imbitrrdi 162	1 |- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E! x ps -> E! x ph ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1371   E.wex 1515   E!weu 2054   E*wmo 2055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058

This theorem is referenced by: (None)