Theorem euim 1419

Description: Add existential uniqueness quantifiers to an implication. Note the reversed implication in the antecedent.

Assertion

Ref	Expression
euim	|- ((E.xph /\ A.x(ph -> ps)) -> (E!xps -> E!xph))

Proof of Theorem euim

Step	Hyp	Ref	Expression
1		euimmo 1418	. . 3 |- (A.x(ph -> ps) -> (E!xps -> E*xph))
2	1	adantl 388	. 2 |- ((E.xph /\ A.x(ph -> ps)) -> (E!xps -> E*xph))
3		exmoeu2 1412	. . 3 |- (E.xph -> (E*xph <-> E!xph))
4	3	adantr 389	. 2 |- ((E.xph /\ A.x(ph -> ps)) -> (E*xph <-> E!xph))
5	2, 4	sylibd 202	1 |- ((E.xph /\ A.x(ph -> ps)) -> (E!xps -> E!xph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952  E.wex 978  E!weu 1378  E*wmo 1379

This theorem is referenced by:  euor2 1435

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381

Copyright terms: Public domain