Theorem bm1.1 1088

Description: Any set which has a property is the only set with that property. Theorem 1.1 of [BellMachover] p. 462.

Hypothesis

Ref	Expression
bm1.1.1	|- (ph -> A.xph)

Assertion

Ref	Expression
bm1.1	|- (E.xA.y(y e. x <-> ph) -> E!xA.y(y e. x <-> ph))

Distinct variable group(s):   x,y

Proof of Theorem bm1.1

Step	Hyp	Ref	Expression
1		19.26 749	. . . . . 6 |- (A.y((y e. x <-> ph) /\ (y e. z <-> ph)) <-> (A.y(y e. x <-> ph) /\ A.y(y e. z <-> ph)))
2		biantr 556	. . . . . . . 8 |- (((y e. x <-> ph) /\ (y e. z <-> ph)) -> (y e. x <-> y e. z))
3	2	19.20i 691	. . . . . . 7 |- (A.y((y e. x <-> ph) /\ (y e. z <-> ph)) -> A.y(y e. x <-> y e. z))
4		ax-ext 1074	. . . . . . 7 |- (A.y(y e. x <-> y e. z) -> x = z)
5	3, 4	syl 12	. . . . . 6 |- (A.y((y e. x <-> ph) /\ (y e. z <-> ph)) -> x = z)
6	1, 5	sylbir 176	. . . . 5 |- ((A.y(y e. x <-> ph) /\ A.y(y e. z <-> ph)) -> x = z)
7		ax-17 925	. . . . . . . 8 |- (y e. z -> A.x y e. z)
8		bm1.1.1	. . . . . . . 8 |- (ph -> A.xph)
9	7, 8	hbbi 705	. . . . . . 7 |- ((y e. z <-> ph) -> A.x(y e. z <-> ph))
10	9	hbal 700	. . . . . 6 |- (A.y(y e. z <-> ph) -> A.xA.y(y e. z <-> ph))
11		a14b 820	. . . . . . . 8 |- (x = z -> (y e. x <-> y e. z))
12	11	bibi1d 471	. . . . . . 7 |- (x = z -> ((y e. x <-> ph) <-> (y e. z <-> ph)))
13	12	bialdv 935	. . . . . 6 |- (x = z -> (A.y(y e. x <-> ph) <-> A.y(y e. z <-> ph)))
14	10, 13	sbie 904	. . . . 5 |- ([z / x]A.y(y e. x <-> ph) <-> A.y(y e. z <-> ph))
15	6, 14	sylan2b 347	. . . 4 |- ((A.y(y e. x <-> ph) /\ [z / x]A.y(y e. x <-> ph)) -> x = z)
16	15	gen2 681	. . 3 |- A.xA.z((A.y(y e. x <-> ph) /\ [z / x]A.y(y e. x <-> ph)) -> x = z)
17	16	jctr 239	. 2 |- (E.xA.y(y e. x <-> ph) -> (E.xA.y(y e. x <-> ph) /\ A.xA.z((A.y(y e. x <-> ph) /\ [z / x]A.y(y e. x <-> ph)) -> x = z)))
18		ax-17 925	. . 3 |- (A.y(y e. x <-> ph) -> A.zA.y(y e. x <-> ph))
19	18	eu2 1023	. 2 |- (E!xA.y(y e. x <-> ph) <-> (E.xA.y(y e. x <-> ph) /\ A.xA.z((A.y(y e. x <-> ph) /\ [z / x]A.y(y e. x <-> ph)) -> x = z)))
20	17, 19	sylibr 175	1 |- (E.xA.y(y e. x <-> ph) -> E!xA.y(y e. x <-> ph))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803  [wsb 852  E!weu 1007

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-14 805  ax-16 922  ax-17 925  ax-ext 1074

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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