Theorem bm1.1 1870

Description: Any set defined by a property is the only set defined by 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:   x,y

Proof of Theorem bm1.1

Step	Hyp	Ref	Expression
1		19.26 1416	. . . . . 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 814	. . . . . . . 8 |- (((y e. x <-> ph) /\ (y e. z <-> ph)) -> (y e. x <-> y e. z))
3	2	alimi 1338	. . . . . . 7 |- (A.y((y e. x <-> ph) /\ (y e. z <-> ph)) -> A.y(y e. x <-> y e. z))
4		ax-ext 1865	. . . . . . 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 218	. . . . 5 |- ((A.y(y e. x <-> ph) /\ A.y(y e. z <-> ph)) -> x = z)
7		ax-17 1317	. . . . . . . 8 |- (y e. z -> A.x y e. z)
8		bm1.1.1	. . . . . . . 8 |- (ph -> A.xph)
9	7, 8	hbbi 1357	. . . . . . 7 |- ((y e. z <-> ph) -> A.x(y e. z <-> ph))
10	9	hbal 1352	. . . . . 6 |- (A.y(y e. z <-> ph) -> A.xA.y(y e. z <-> ph))
11		elequ2 1497	. . . . . . . 8 |- (x = z -> (y e. x <-> y e. z))
12	11	bibi1d 681	. . . . . . 7 |- (x = z -> ((y e. x <-> ph) <-> (y e. z <-> ph)))
13	12	albidv 1656	. . . . . 6 |- (x = z -> (A.y(y e. x <-> ph) <-> A.y(y e. z <-> ph)))
14	10, 13	sbie 1565	. . . . 5 |- ([z / x]A.y(y e. x <-> ph) <-> A.y(y e. z <-> ph))
15	6, 14	sylan2b 501	. . . 4 |- ((A.y(y e. x <-> ph) /\ [z / x]A.y(y e. x <-> ph)) -> x = z)
16	15	gen2 1329	. . 3 |- A.xA.z((A.y(y e. x <-> ph) /\ [z / x]A.y(y e. x <-> ph)) -> x = z)
17	16	jctr 315	. 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 1317	. . 3 |- (A.y(y e. x <-> ph) -> A.zA.y(y e. x <-> ph))
19	18	eu2 1791	. 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 217	1 |- (E.xA.y(y e. x <-> ph) -> E!xA.y(y e. x <-> ph))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  E!weu 1771

This theorem is referenced by:  zfnuleu 3442

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775

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