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Theorem bm1.1 2073
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 1466 . . . . . . . 8 𝑥 𝑦𝑧
2 bm1.1.1 . . . . . . . 8 𝑥𝜑
31, 2nfbi 1526 . . . . . . 7 𝑥(𝑦𝑧𝜑)
43nfal 1513 . . . . . 6 𝑥𝑦(𝑦𝑧𝜑)
5 elequ2 1648 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
65bibi1d 231 . . . . . . 7 (𝑥 = 𝑧 → ((𝑦𝑥𝜑) ↔ (𝑦𝑧𝜑)))
76albidv 1752 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦(𝑦𝑥𝜑) ↔ ∀𝑦(𝑦𝑧𝜑)))
84, 7sbie 1721 . . . . 5 ([𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑) ↔ ∀𝑦(𝑦𝑧𝜑))
9 19.26 1415 . . . . . 6 (∀𝑦((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) ↔ (∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)))
10 biantr 898 . . . . . . . 8 (((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) → (𝑦𝑥𝑦𝑧))
1110alimi 1389 . . . . . . 7 (∀𝑦((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) → ∀𝑦(𝑦𝑥𝑦𝑧))
12 ax-ext 2070 . . . . . . 7 (∀𝑦(𝑦𝑥𝑦𝑧) → 𝑥 = 𝑧)
1311, 12syl 14 . . . . . 6 (∀𝑦((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) → 𝑥 = 𝑧)
149, 13sylbir 133 . . . . 5 ((∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)) → 𝑥 = 𝑧)
158, 14sylan2b 281 . . . 4 ((∀𝑦(𝑦𝑥𝜑) ∧ [𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑)) → 𝑥 = 𝑧)
1615gen2 1384 . . 3 𝑥𝑧((∀𝑦(𝑦𝑥𝜑) ∧ [𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑)) → 𝑥 = 𝑧)
1716jctr 308 . 2 (∃𝑥𝑦(𝑦𝑥𝜑) → (∃𝑥𝑦(𝑦𝑥𝜑) ∧ ∀𝑥𝑧((∀𝑦(𝑦𝑥𝜑) ∧ [𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑)) → 𝑥 = 𝑧)))
18 nfv 1466 . . 3 𝑧𝑦(𝑦𝑥𝜑)
1918eu2 1992 . 2 (∃!𝑥𝑦(𝑦𝑥𝜑) ↔ (∃𝑥𝑦(𝑦𝑥𝜑) ∧ ∀𝑥𝑧((∀𝑦(𝑦𝑥𝜑) ∧ [𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑)) → 𝑥 = 𝑧)))
2017, 19sylibr 132 1 (∃𝑥𝑦(𝑦𝑥𝜑) → ∃!𝑥𝑦(𝑦𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287  wnf 1394  wex 1426  [wsb 1692  ∃!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951
This theorem is referenced by:  zfnuleu  3955
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