Theorem eu1 1019

Description: An alternate way of expressing 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(s):   x,y

Proof of Theorem eu1

Step	Hyp	Ref	Expression
1		hbs1 986	. . 3 |- ([y / x]ph -> A.x[y / x]ph)
2	1	euf 1011	. 2 |- (E!y[y / x]ph <-> E.xA.y([y / x]ph <-> y = x))
3		eu1.1	. . 3 |- (ph -> A.yph)
4	3	sb8eu 1017	. 2 |- (E!xph <-> E!y[y / x]ph)
5		eqcomb 812	. . . . . . 7 |- (x = y <-> y = x)
6	5	imbi2i 160	. . . . . 6 |- (([y / x]ph -> x = y) <-> ([y / x]ph -> y = x))
7	6	bial 695	. . . . 5 |- (A.y([y / x]ph -> x = y) <-> A.y([y / x]ph -> y = x))
8	3	sb5f1 917	. . . . 5 |- (ph <-> A.y(y = x -> [y / x]ph))
9	7, 8	anbi12i 369	. . . 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 333	. . . 4 |- ((ph /\ A.y([y / x]ph -> x = y)) <-> (A.y([y / x]ph -> x = y) /\ ph))
11		albi 785	. . . 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	3bitr4 158	. . 3 |- ((ph /\ A.y([y / x]ph -> x = y)) <-> A.y([y / x]ph <-> y = x))
13	12	biex 733	. 2 |- (E.x(ph /\ A.y([y / x]ph -> x = y)) <-> E.xA.y([y / x]ph <-> y = x))
14	2, 4, 13	3bitr4 158	1 |- (E!xph <-> E.x(ph /\ A.y([y / x]ph -> x = y)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797  [wsb 852  E!weu 1007

This theorem is referenced by:  euex 1021  eu2 1023  kmlem15 3594

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-ex 679  df-sb 853  df-eu 1009

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