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Theorem bm1.1 2338
 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 1619 . . . . . . . 8 x y z
2 bm1.1.1 . . . . . . . 8 xφ
31, 2nfbi 1834 . . . . . . 7 x(y zφ)
43nfal 1842 . . . . . 6 xy(y zφ)
5 elequ2 1715 . . . . . . . 8 (x = z → (y xy z))
65bibi1d 310 . . . . . . 7 (x = z → ((y xφ) ↔ (y zφ)))
76albidv 1625 . . . . . 6 (x = z → (y(y xφ) ↔ y(y zφ)))
84, 7sbie 2038 . . . . 5 ([z / x]y(y xφ) ↔ y(y zφ))
9 19.26 1593 . . . . . 6 (y((y xφ) (y zφ)) ↔ (y(y xφ) y(y zφ)))
10 biantr 897 . . . . . . . 8 (((y xφ) (y zφ)) → (y xy z))
1110alimi 1559 . . . . . . 7 (y((y xφ) (y zφ)) → y(y xy z))
12 ax-ext 2334 . . . . . . 7 (y(y xy z) → x = z)
1311, 12syl 15 . . . . . 6 (y((y xφ) (y zφ)) → x = z)
149, 13sylbir 204 . . . . 5 ((y(y xφ) y(y zφ)) → x = z)
158, 14sylan2b 461 . . . 4 ((y(y xφ) [z / x]y(y xφ)) → x = z)
1615gen2 1547 . . 3 xz((y(y xφ) [z / x]y(y xφ)) → x = z)
1716jctr 526 . 2 (xy(y xφ) → (xy(y xφ) xz((y(y xφ) [z / x]y(y xφ)) → x = z)))
18 nfv 1619 . . 3 zy(y xφ)
1918eu2 2229 . 2 (∃!xy(y xφ) ↔ (xy(y xφ) xz((y(y xφ) [z / x]y(y xφ)) → x = z)))
2017, 19sylibr 203 1 (xy(y xφ) → ∃!xy(y xφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∃!weu 2204 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208 This theorem is referenced by: (None)
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