Theorem euim 2523

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. 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 2522	. . 3 |- ( A. x ( ph -> ps ) -> ( E! x ps -> E* x ph ) )
3	1, 2	anim12ii 594	. 2 |- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E! x ps -> ( E. x ph /\ E* x ph ) ) )
4		eu5 2496	. 2 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
5	3, 4	syl6ibr 242	1 |- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E! x ps -> E! x ph ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   E!weu 2470   E*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  2eu1  2553