Theorem eu2 1396

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 1394	. . 3 |- (E!xph -> E.xph)
2		eu2.1	. . . . 5 |- (ph -> A.yph)
3	2	eumo0 1395	. . . 4 |- (E!xph -> E.yA.x(ph -> x = y))
4	2	mo 1393	. . . 4 |- (E.yA.x(ph -> x = y) <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
5	3, 4	sylib 198	. . 3 |- (E!xph -> A.xA.y((ph /\ [y / x]ph) -> x = y))
6	1, 5	jca 288	. 2 |- (E!xph -> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))
7		19.29r 1072	. . . 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 347	. . . . . . . . 9 |- (((ph /\ [y / x]ph) -> x = y) <-> (ph -> ([y / x]ph -> x = y)))
9	8	albii 999	. . . . . . . 8 |- (A.y((ph /\ [y / x]ph) -> x = y) <-> A.y(ph -> ([y / x]ph -> x = y)))
10	2	19.21 1056	. . . . . . . 8 |- (A.y(ph -> ([y / x]ph -> x = y)) <-> (ph -> A.y([y / x]ph -> x = y)))
11	9, 10	bitr 173	. . . . . . 7 |- (A.y((ph /\ [y / x]ph) -> x = y) <-> (ph -> A.y([y / x]ph -> x = y)))
12	11	anbi2i 480	. . . . . 6 |- ((ph /\ A.y((ph /\ [y / x]ph) -> x = y)) <-> (ph /\ (ph -> A.y([y / x]ph -> x = y))))
13		abai 479	. . . . . 6 |- ((ph /\ A.y([y / x]ph -> x = y)) <-> (ph /\ (ph -> A.y([y / x]ph -> x = y))))
14	12, 13	bitr4 176	. . . . 5 |- ((ph /\ A.y((ph /\ [y / x]ph) -> x = y)) <-> (ph /\ A.y([y / x]ph -> x = y)))
15	14	exbii 1051	. . . 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 198	. . 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 1392	. . 3 |- (E!xph <-> E.x(ph /\ A.y([y / x]ph -> x = y)))
18	16, 17	sylibr 200	. 2 |- ((E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)) -> E!xph)
19	6, 18	impbi 157	1 |- (E!xph <-> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956  E.wex 980  [wsbc 1170  E!weu 1380

This theorem is referenced by:  eu3 1397  bm1.1 1462  reu2 1930

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-ex 981  df-sb 1172  df-eu 1382

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