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Theorem bm1.1 2011
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 1408 . . . . . . . 8 x y z
2 bm1.1.1 . . . . . . . 8 xφ
31, 2nfbi 1470 . . . . . . 7 x(y zφ)
43nfal 1457 . . . . . 6 xy(y zφ)
5 elequ2 1589 . . . . . . . 8 (x = z → (y xy z))
65bibi1d 222 . . . . . . 7 (x = z → ((y xφ) ↔ (y zφ)))
76albidv 1693 . . . . . 6 (x = z → (y(y xφ) ↔ y(y zφ)))
84, 7sbie 1662 . . . . 5 ([z / x]y(y xφ) ↔ y(y zφ))
9 19.26 1355 . . . . . 6 (y((y xφ) (y zφ)) ↔ (y(y xφ) y(y zφ)))
10 biantr 852 . . . . . . . 8 (((y xφ) (y zφ)) → (y xy z))
1110alimi 1329 . . . . . . 7 (y((y xφ) (y zφ)) → y(y xy z))
12 ax-ext 2008 . . . . . . 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 1324 . . 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 1408 . . 3 zy(y xφ)
1918eu2 1930 . 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 1231  wnf 1334  wex 1368  [wsb 1633  ∃!weu 1886
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 1321  ax-7 1322  ax-gen 1323  ax-ie1 1369  ax-ie2 1370  ax-8 1382  ax-10 1383  ax-11 1384  ax-i12 1385  ax-bnd 1386  ax-4 1387  ax-14 1392  ax-17 1406  ax-i9 1410  ax-ial 1415  ax-i5r 1416  ax-ext 2008
This theorem depends on definitions:  df-bi 110  df-tru 1236  df-nf 1335  df-sb 1634  df-eu 1889
This theorem is referenced by:  zfnuleu  3857
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