Theorem bm1.1 2338

Description: Any set defined by a property is the only set defined by that property. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.)

Hypothesis

Ref	Expression
bm1.1.1	|- F/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

Allowed substitution hints:   ph(x,y)

Proof of Theorem bm1.1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . . . . . 8 |- F/x y e. z
2		bm1.1.1	. . . . . . . 8 |- F/xph
3	1, 2	nfbi 1834	. . . . . . 7 |- F/x(y e. z <-> ph)
4	3	nfal 1842	. . . . . 6 |- F/xA.y(y e. z <-> ph)
5		elequ2 1715	. . . . . . . 8 |- (x = z -> (y e. x <-> y e. z))
6	5	bibi1d 310	. . . . . . 7 |- (x = z -> ((y e. x <-> ph) <-> (y e. z <-> ph)))
7	6	albidv 1625	. . . . . 6 |- (x = z -> (A.y(y e. x <-> ph) <-> A.y(y e. z <-> ph)))
8	4, 7	sbie 2038	. . . . 5 |- ([z / x]A.y(y e. x <-> ph) <-> A.y(y e. z <-> ph))
9		19.26 1593	. . . . . 6 |- (A.y((y e. x <-> ph) /\ (y e. z <-> ph)) <-> (A.y(y e. x <-> ph) /\ A.y(y e. z <-> ph)))
10		biantr 897	. . . . . . . 8 |- (((y e. x <-> ph) /\ (y e. z <-> ph)) -> (y e. x <-> y e. z))
11	10	alimi 1559	. . . . . . 7 |- (A.y((y e. x <-> ph) /\ (y e. z <-> ph)) -> A.y(y e. x <-> y e. z))
12		ax-ext 2334	. . . . . . 7 |- (A.y(y e. x <-> y e. z) -> x = z)
13	11, 12	syl 15	. . . . . 6 |- (A.y((y e. x <-> ph) /\ (y e. z <-> ph)) -> x = z)
14	9, 13	sylbir 204	. . . . 5 |- ((A.y(y e. x <-> ph) /\ A.y(y e. z <-> ph)) -> x = z)
15	8, 14	sylan2b 461	. . . 4 |- ((A.y(y e. x <-> ph) /\ [z / x]A.y(y e. x <-> ph)) -> x = z)
16	15	gen2 1547	. . 3 |- A.xA.z((A.y(y e. x <-> ph) /\ [z / x]A.y(y e. x <-> ph)) -> x = z)
17	16	jctr 526	. 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		nfv 1619	. . 3 |- F/zA.y(y e. x <-> ph)
19	18	eu2 2229	. 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 203	1 |- (E.xA.y(y e. x <-> ph) -> E!xA.y(y e. x <-> ph))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   F/wnf 1544   = wceq 1642  [wsb 1648   e. wcel 1710  E!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208

This theorem is referenced by: (None)