Theorem eupickb 1056

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 1055	. . 3 |- ((E!xph /\ E.x(ph /\ ps)) -> (ph -> ps))
2	1	3adant2 598	. 2 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ph -> ps))
3		3simpc 593	. . 3 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (E!xps /\ E.x(ph /\ ps)))
4		ancom 333	. . . . . 6 |- ((ph /\ ps) <-> (ps /\ ph))
5	4	biimp 133	. . . . 5 |- ((ph /\ ps) -> (ps /\ ph))
6	5	19.22i 723	. . . 4 |- (E.x(ph /\ ps) -> E.x(ps /\ ph))
7	6	anim2i 270	. . 3 |- ((E!xps /\ E.x(ph /\ ps)) -> (E!xps /\ E.x(ps /\ ph)))
8		eupick 1055	. . 3 |- ((E!xps /\ E.x(ps /\ ph)) -> (ps -> ph))
9	3, 7, 8	3syl 21	. 2 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ps -> ph))
10	2, 9	impbid 397	1 |- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ph <-> ps))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   /\ w3a 581  E.wex 678  E!weu 1007

This theorem is referenced by:  euuni 1954

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

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