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Theorem bm1.1 2122
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(𝑦𝑥𝜑) → ∃!𝑥𝑦(𝑦𝑥𝜑))
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 1508 . . . . . . . 8 𝑥 𝑦𝑧
2 bm1.1.1 . . . . . . . 8 𝑥𝜑
31, 2nfbi 1568 . . . . . . 7 𝑥(𝑦𝑧𝜑)
43nfal 1555 . . . . . 6 𝑥𝑦(𝑦𝑧𝜑)
5 elequ2 1691 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
65bibi1d 232 . . . . . . 7 (𝑥 = 𝑧 → ((𝑦𝑥𝜑) ↔ (𝑦𝑧𝜑)))
76albidv 1796 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦(𝑦𝑥𝜑) ↔ ∀𝑦(𝑦𝑧𝜑)))
84, 7sbie 1764 . . . . 5 ([𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑) ↔ ∀𝑦(𝑦𝑧𝜑))
9 19.26 1457 . . . . . 6 (∀𝑦((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) ↔ (∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)))
10 biantr 936 . . . . . . . 8 (((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) → (𝑦𝑥𝑦𝑧))
1110alimi 1431 . . . . . . 7 (∀𝑦((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) → ∀𝑦(𝑦𝑥𝑦𝑧))
12 ax-ext 2119 . . . . . . 7 (∀𝑦(𝑦𝑥𝑦𝑧) → 𝑥 = 𝑧)
1311, 12syl 14 . . . . . 6 (∀𝑦((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) → 𝑥 = 𝑧)
149, 13sylbir 134 . . . . 5 ((∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)) → 𝑥 = 𝑧)
158, 14sylan2b 285 . . . 4 ((∀𝑦(𝑦𝑥𝜑) ∧ [𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑)) → 𝑥 = 𝑧)
1615gen2 1426 . . 3 𝑥𝑧((∀𝑦(𝑦𝑥𝜑) ∧ [𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑)) → 𝑥 = 𝑧)
1716jctr 313 . 2 (∃𝑥𝑦(𝑦𝑥𝜑) → (∃𝑥𝑦(𝑦𝑥𝜑) ∧ ∀𝑥𝑧((∀𝑦(𝑦𝑥𝜑) ∧ [𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑)) → 𝑥 = 𝑧)))
18 nfv 1508 . . 3 𝑧𝑦(𝑦𝑥𝜑)
1918eu2 2041 . 2 (∃!𝑥𝑦(𝑦𝑥𝜑) ↔ (∃𝑥𝑦(𝑦𝑥𝜑) ∧ ∀𝑥𝑧((∀𝑦(𝑦𝑥𝜑) ∧ [𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑)) → 𝑥 = 𝑧)))
2017, 19sylibr 133 1 (∃𝑥𝑦(𝑦𝑥𝜑) → ∃!𝑥𝑦(𝑦𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1329  wnf 1436  wex 1468  [wsb 1735  ∃!weu 1997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000
This theorem is referenced by:  zfnuleu  4047
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