Theorem eupickb 1475

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 1473	. . 3 |- ((E!xph /\ E.x(ph /\ ps)) -> (ph -> ps))
2	1	3adant2 804	. 2 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ph -> ps))
3		3simpc 793	. . 3 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (E!xps /\ E.x(ph /\ ps)))
4		pm3.22 440	. . . . 5 |- ((ph /\ ps) -> (ps /\ ph))
5	4	19.22i 1076	. . . 4 |- (E.x(ph /\ ps) -> E.x(ps /\ ph))
6	5	anim2i 333	. . 3 |- ((E!xps /\ E.x(ph /\ ps)) -> (E!xps /\ E.x(ps /\ ph)))
7		eupick 1473	. . 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 519	1 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ph <-> ps))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   /\ w3a 781  E.wex 1016  E!weu 1419

This theorem is referenced by:  euuni 3105

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422

Copyright terms: Public domain