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Theorem bm1.1 2007
 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 xφ
Assertion
Ref Expression
bm1.1 (xy(y xφ) → ∃!xy(y xφ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem bm1.1
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfv 1402 . . . . . . . 8 x y z
2 bm1.1.1 . . . . . . . 8 xφ
31, 2nfbi 1463 . . . . . . 7 x(y zφ)
43nfal 1450 . . . . . 6 xy(y zφ)
5 elequ2 1583 . . . . . . . 8 (x = z → (y xy z))
65bibi1d 222 . . . . . . 7 (x = z → ((y xφ) ↔ (y zφ)))
76albidv 1687 . . . . . 6 (x = z → (y(y xφ) ↔ y(y zφ)))
84, 7sbie 1656 . . . . 5 ([z / x]y(y xφ) ↔ y(y zφ))
9 19.26 1350 . . . . . 6 (y((y xφ) (y zφ)) ↔ (y(y xφ) y(y zφ)))
10 biantr 847 . . . . . . . 8 (((y xφ) (y zφ)) → (y xy z))
1110alimi 1324 . . . . . . 7 (y((y xφ) (y zφ)) → y(y xy z))
12 ax-ext 2004 . . . . . . 7 (y(y xy z) → x = z)
1311, 12syl 14 . . . . . 6 (y((y xφ) (y zφ)) → x = z)
149, 13sylbir 125 . . . . 5 ((y(y xφ) y(y zφ)) → x = z)
158, 14sylan2b 271 . . . 4 ((y(y xφ) [z / x]y(y xφ)) → x = z)
1615gen2 1319 . . 3 xz((y(y xφ) [z / x]y(y xφ)) → x = z)
1716jctr 298 . 2 (xy(y xφ) → (xy(y xφ) xz((y(y xφ) [z / x]y(y xφ)) → x = z)))
18 nfv 1402 . . 3 zy(y xφ)
1918eu2 1926 . 2 (∃!xy(y xφ) ↔ (xy(y xφ) xz((y(y xφ) [z / x]y(y xφ)) → x = z)))
2017, 19sylibr 137 1 (xy(y xφ) → ∃!xy(y xφ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1226  Ⅎwnf 1329  ∃wex 1362  [wsb 1627  ∃!weu 1882 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885 This theorem is referenced by:  zfnuleu  3855
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