Theorem eu2 1435

Description: An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26.

Hypothesis

Ref	Expression
eu2.1	|- (ph -> A.yph)

Assertion

Ref	Expression
eu2	|- (E!xph <-> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))

Distinct variable group:   x,y

Proof of Theorem eu2

Step	Hyp	Ref	Expression
1		euex 1433	. . 3 |- (E!xph -> E.xph)
2		eu2.1	. . . . 5 |- (ph -> A.yph)
3	2	eumo0 1434	. . . 4 |- (E!xph -> E.yA.x(ph -> x = y))
4	2	mo 1432	. . . 4 |- (E.yA.x(ph -> x = y) <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
5	3, 4	sylib 196	. . 3 |- (E!xph -> A.xA.y((ph /\ [y / x]ph) -> x = y))
6	1, 5	jca 286	. 2 |- (E!xph -> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))
7		19.29r 1108	. . . 4 |- ((E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)) -> E.x(ph /\ A.y((ph /\ [y / x]ph) -> x = y)))
8		impexp 345	. . . . . . . . 9 |- (((ph /\ [y / x]ph) -> x = y) <-> (ph -> ([y / x]ph -> x = y)))
9	8	albii 1035	. . . . . . . 8 |- (A.y((ph /\ [y / x]ph) -> x = y) <-> A.y(ph -> ([y / x]ph -> x = y)))
10	2	19.21 1092	. . . . . . . 8 |- (A.y(ph -> ([y / x]ph -> x = y)) <-> (ph -> A.y([y / x]ph -> x = y)))
11	9, 10	bitri 171	. . . . . . 7 |- (A.y((ph /\ [y / x]ph) -> x = y) <-> (ph -> A.y([y / x]ph -> x = y)))
12	11	anbi2i 483	. . . . . 6 |- ((ph /\ A.y((ph /\ [y / x]ph) -> x = y)) <-> (ph /\ (ph -> A.y([y / x]ph -> x = y))))
13		abai 482	. . . . . 6 |- ((ph /\ A.y([y / x]ph -> x = y)) <-> (ph /\ (ph -> A.y([y / x]ph -> x = y))))
14	12, 13	bitr4i 174	. . . . 5 |- ((ph /\ A.y((ph /\ [y / x]ph) -> x = y)) <-> (ph /\ A.y([y / x]ph -> x = y)))
15	14	exbii 1087	. . . 4 |- (E.x(ph /\ A.y((ph /\ [y / x]ph) -> x = y)) <-> E.x(ph /\ A.y([y / x]ph -> x = y)))
16	7, 15	sylib 196	. . 3 |- ((E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)) -> E.x(ph /\ A.y([y / x]ph -> x = y)))
17	2	eu1 1431	. . 3 |- (E!xph <-> E.x(ph /\ A.y([y / x]ph -> x = y)))
18	16, 17	sylibr 198	. 2 |- ((E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)) -> E!xph)
19	6, 18	impbii 155	1 |- (E!xph <-> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221  A.wal 990   = wceq 992  E.wex 1016  [wsbc 1207  E!weu 1419

This theorem is referenced by:  eu3 1436  bm1.1 1504  reu2 1976

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-ex 1017  df-sb 1209  df-eu 1421

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