Theorem eu1 1431

Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110.

Hypothesis

Ref	Expression
eu1.1	|- (ph -> A.yph)

Assertion

Ref	Expression
eu1	|- (E!xph <-> E.x(ph /\ A.y([y / x]ph -> x = y)))

Distinct variable group:   x,y

Proof of Theorem eu1

Step	Hyp	Ref	Expression
1		hbs1 1371	. . 3 |- ([y / x]ph -> A.x[y / x]ph)
2	1	euf 1423	. 2 |- (E!y[y / x]ph <-> E.xA.y([y / x]ph <-> y = x))
3		eu1.1	. . 3 |- (ph -> A.yph)
4	3	sb8eu 1429	. 2 |- (E!xph <-> E!y[y / x]ph)
5		equcom 1166	. . . . . . 7 |- (x = y <-> y = x)
6	5	imbi2i 183	. . . . . 6 |- (([y / x]ph -> x = y) <-> ([y / x]ph -> y = x))
7	6	albii 1035	. . . . 5 |- (A.y([y / x]ph -> x = y) <-> A.y([y / x]ph -> y = x))
8	3	sb6rf 1298	. . . . 5 |- (ph <-> A.y(y = x -> [y / x]ph))
9	7, 8	anbi12i 485	. . . 4 |- ((A.y([y / x]ph -> x = y) /\ ph) <-> (A.y([y / x]ph -> y = x) /\ A.y(y = x -> [y / x]ph)))
10		ancom 437	. . . 4 |- ((ph /\ A.y([y / x]ph -> x = y)) <-> (A.y([y / x]ph -> x = y) /\ ph))
11		albi 1143	. . . 4 |- (A.y([y / x]ph <-> y = x) <-> (A.y([y / x]ph -> y = x) /\ A.y(y = x -> [y / x]ph)))
12	9, 10, 11	3bitr4i 181	. . 3 |- ((ph /\ A.y([y / x]ph -> x = y)) <-> A.y([y / x]ph <-> y = x))
13	12	exbii 1087	. 2 |- (E.x(ph /\ A.y([y / x]ph -> x = y)) <-> E.xA.y([y / x]ph <-> y = x))
14	2, 4, 13	3bitr4i 181	1 |- (E!xph <-> E.x(ph /\ A.y([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:  euex 1433  eu2 1435  kmlem15 4925

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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