Theorem bm1.1 2440

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.) (Proof shortened by Wolf Lammen, 13-Nov-2019.)

Hypothesis

Ref	Expression
bm1.1.1

Assertion

Ref	Expression
bm1.1

Distinct variable group:   ,

Proof of Theorem bm1.1

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		biantr 931	. . . . 5
2	1	alanimi 1637	. . . 4
3		ax-ext 2435	. . . 4
4	2, 3	syl 16	. . 3
5	4	gen2 1619	. 2
6		nfv 1707	. . . . . 6
7		bm1.1.1	. . . . . 6
8	6, 7	nfbi 1934	. . . . 5
9	8	nfal 1947	. . . 4
10		elequ2 1823	. . . . . 6
11	10	bibi1d 319	. . . . 5
12	11	albidv 1713	. . . 4
13	9, 12	mo4f 2336	. . 3
14		df-mo 2287	. . 3
15	13, 14	bitr3i 251	. 2
16	5, 15	mpbi 208	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  F/wnf 1616  E!weu 2282  E*wmo 2283

This theorem is referenced by:  zfnuleu  4578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287