Theorem bm1.1 1462

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 1067	. . . . . 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 742	. . . . . . . 8 |- (((y e. x <-> ph) /\ (y e. z <-> ph)) -> (y e. x <-> y e. z))
3	2	19.20i 992	. . . . . . 7 |- (A.y((y e. x <-> ph) /\ (y e. z <-> ph)) -> A.y(y e. x <-> y e. z))
4		ax-ext 1459	. . . . . . 7 |- (A.y(y e. x <-> y e. z) -> x = z)
5	3, 4	syl 10	. . . . . 6 |- (A.y((y e. x <-> ph) /\ (y e. z <-> ph)) -> x = z)
6	1, 5	sylbir 201	. . . . 5 |- ((A.y(y e. x <-> ph) /\ A.y(y e. z <-> ph)) -> x = z)
7		ax-17 971	. . . . . . . 8 |- (y e. z -> A.x y e. z)
8		bm1.1.1	. . . . . . . 8 |- (ph -> A.xph)
9	7, 8	hbbi 1010	. . . . . . 7 |- ((y e. z <-> ph) -> A.x(y e. z <-> ph))
10	9	hbal 1005	. . . . . 6 |- (A.y(y e. z <-> ph) -> A.xA.y(y e. z <-> ph))
11		elequ2 1137	. . . . . . . 8 |- (x = z -> (y e. x <-> y e. z))
12	11	bibi1d 619	. . . . . . 7 |- (x = z -> ((y e. x <-> ph) <-> (y e. z <-> ph)))
13	12	albidv 1278	. . . . . 6 |- (x = z -> (A.y(y e. x <-> ph) <-> A.y(y e. z <-> ph)))
14	10, 13	sbie 1196	. . . . 5 |- ([z / x]A.y(y e. x <-> ph) <-> A.y(y e. z <-> ph))
15	6, 14	sylan2b 452	. . . 4 |- ((A.y(y e. x <-> ph) /\ [z / x]A.y(y e. x <-> ph)) -> x = z)
16	15	gen2 983	. . 3 |- A.xA.z((A.y(y e. x <-> ph) /\ [z / x]A.y(y e. x <-> ph)) -> x = z)
17	16	jctr 291	. 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-17 971	. . 3 |- (A.y(y e. x <-> ph) -> A.zA.y(y e. x <-> ph))
19	18	eu2 1396	. 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 200	1 |- (E.xA.y(y e. x <-> ph) -> E!xA.y(y e. x <-> ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  [wsbc 1170  E!weu 1380

This theorem is referenced by:  zfnuleu 2705

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-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

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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