Theorem eupickb 1433

Description: Existential uniqueness "pick" showing wff equivalence.

Assertion

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

Proof of Theorem eupickb

Step	Hyp	Ref	Expression
1		eupick 1432	. . 3 |- ((E!xph /\ E.x(ph /\ ps)) -> (ph -> ps))
2	1	3adant2 797	. 2 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ph -> ps))
3		3simpc 786	. . 3 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (E!xps /\ E.x(ph /\ ps)))
4		pm3.22 438	. . . . 5 |- ((ph /\ ps) -> (ps /\ ph))
5	4	19.22i 1038	. . . 4 |- (E.x(ph /\ ps) -> E.x(ps /\ ph))
6	5	anim2i 335	. . 3 |- ((E!xps /\ E.x(ph /\ ps)) -> (E!xps /\ E.x(ps /\ ph)))
7		eupick 1432	. . 3 |- ((E!xps /\ E.x(ps /\ ph)) -> (ps -> ph))
8	3, 6, 7	3syl 20	. 2 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ps -> ph))
9	2, 8	impbid 515	1 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ph <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774  E.wex 978  E!weu 1378

This theorem is referenced by:  euuni 2876

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-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381

Copyright terms: Public domain