Theorem eupickb 1836

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 1834	. . 3 |- ((E!xph /\ E.x(ph /\ ps)) -> (ph -> ps))
2	1	3adant2 895	. 2 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ph -> ps))
3		3simpc 874	. . 3 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (E!xps /\ E.x(ph /\ ps)))
4		pm3.22 486	. . . . 5 |- ((ph /\ ps) -> (ps /\ ph))
5	4	eximi 1387	. . . 4 |- (E.x(ph /\ ps) -> E.x(ps /\ ph))
6	5	anim2i 362	. . 3 |- ((E!xps /\ E.x(ph /\ ps)) -> (E!xps /\ E.x(ps /\ ph)))
7		eupick 1834	. . 3 |- ((E!xps /\ E.x(ps /\ ph)) -> (ps -> ph))
8	3, 6, 7	3syl 24	. 2 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ps -> ph))
9	2, 8	impbid 574	1 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ph <-> ps))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  E.wex 1326  E!weu 1771

This theorem is referenced by:  euuni 3807

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776

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