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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 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 1418 . . . . . . . 8 x y z
2 bm1.1.1 . . . . . . . 8 xφ
31, 2nfbi 1478 . . . . . . 7 x(y zφ)
43nfal 1465 . . . . . 6 xy(y zφ)
5 elequ2 1598 . . . . . . . 8 (x = z → (y xy z))
65bibi1d 222 . . . . . . 7 (x = z → ((y xφ) ↔ (y zφ)))
76albidv 1702 . . . . . 6 (x = z → (y(y xφ) ↔ y(y zφ)))
84, 7sbie 1671 . . . . 5 ([z / x]y(y xφ) ↔ y(y zφ))
9 19.26 1367 . . . . . 6 (y((y xφ) (y zφ)) ↔ (y(y xφ) y(y zφ)))
10 biantr 858 . . . . . . . 8 (((y xφ) (y zφ)) → (y xy z))
1110alimi 1341 . . . . . . 7 (y((y xφ) (y zφ)) → y(y xy z))
12 ax-ext 2019 . . . . . . 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 1336 . . 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 1418 . . 3 zy(y xφ)
1918eu2 1941 . 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 1240  wnf 1346  wex 1378  [wsb 1642  ∃!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
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