Theorem bm1.1 2022

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 1418	. . . . . . . 8 |- F/x y e. z
2		bm1.1.1	. . . . . . . 8 |- F/xph
3	1, 2	nfbi 1478	. . . . . . 7 |- F/x(y e. z <-> ph)
4	3	nfal 1465	. . . . . 6 |- F/xA.y(y e. z <-> ph)
5		elequ2 1598	. . . . . . . 8 |- (x = z -> (y e. x <-> y e. z))
6	5	bibi1d 222	. . . . . . 7 |- (x = z -> ((y e. x <-> ph) <-> (y e. z <-> ph)))
7	6	albidv 1702	. . . . . 6 |- (x = z -> (A.y(y e. x <-> ph) <-> A.y(y e. z <-> ph)))
8	4, 7	sbie 1671	. . . . 5 |- ([z / x]A.y(y e. x <-> ph) <-> A.y(y e. z <-> ph))
9		19.26 1367	. . . . . 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 858	. . . . . . . 8 |- (((y e. x <-> ph) /\ (y e. z <-> ph)) -> (y e. x <-> y e. z))
11	10	alimi 1341	. . . . . . 7 |- (A.y((y e. x <-> ph) /\ (y e. z <-> ph)) -> A.y(y e. x <-> y e. z))
12		ax-ext 2019	. . . . . . 7 |- (A.y(y e. x <-> y e. z) -> x = z)
13	11, 12	syl 14	. . . . . 6 |- (A.y((y e. x <-> ph) /\ (y e. z <-> ph)) -> x = z)
14	9, 13	sylbir 125	. . . . 5 |- ((A.y(y e. x <-> ph) /\ A.y(y e. z <-> ph)) -> x = z)
15	8, 14	sylan2b 271	. . . 4 |- ((A.y(y e. x <-> ph) /\ [z / x]A.y(y e. x <-> ph)) -> x = z)
16	15	gen2 1336	. . 3 |- A.xA.z((A.y(y e. x <-> ph) /\ [z / x]A.y(y e. x <-> ph)) -> x = z)
17	16	jctr 298	. 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 1418	. . 3 |- F/zA.y(y e. x <-> ph)
19	18	eu2 1941	. 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 137	1 |- (E.xA.y(y e. x <-> ph) -> E!xA.y(y e. x <-> ph))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240   F/wnf 1346  E.wex 1378  [wsb 1642  E!weu 1897

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900

This theorem is referenced by:  zfnuleu  3872