Theorem eupickb 1435

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 1434	. . 3 |- ((E!xph /\ E.x(ph /\ ps)) -> (ph -> ps))
2	1	3adant2 798	. 2 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ph -> ps))
3		3simpc 787	. . 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 1040	. . . 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 1434	. . 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 516	1 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ph <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775  E.wex 980  E!weu 1380

This theorem is referenced by:  euuni 2881

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383

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