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Theorem bm1.1 2041
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 1437 . . . . . . . 8 𝑥 𝑦𝑧
2 bm1.1.1 . . . . . . . 8 𝑥𝜑
31, 2nfbi 1497 . . . . . . 7 𝑥(𝑦𝑧𝜑)
43nfal 1484 . . . . . 6 𝑥𝑦(𝑦𝑧𝜑)
5 elequ2 1617 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
65bibi1d 226 . . . . . . 7 (𝑥 = 𝑧 → ((𝑦𝑥𝜑) ↔ (𝑦𝑧𝜑)))
76albidv 1721 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦(𝑦𝑥𝜑) ↔ ∀𝑦(𝑦𝑧𝜑)))
84, 7sbie 1690 . . . . 5 ([𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑) ↔ ∀𝑦(𝑦𝑧𝜑))
9 19.26 1386 . . . . . 6 (∀𝑦((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) ↔ (∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)))
10 biantr 870 . . . . . . . 8 (((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) → (𝑦𝑥𝑦𝑧))
1110alimi 1360 . . . . . . 7 (∀𝑦((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) → ∀𝑦(𝑦𝑥𝑦𝑧))
12 ax-ext 2038 . . . . . . 7 (∀𝑦(𝑦𝑥𝑦𝑧) → 𝑥 = 𝑧)
1311, 12syl 14 . . . . . 6 (∀𝑦((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) → 𝑥 = 𝑧)
149, 13sylbir 129 . . . . 5 ((∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)) → 𝑥 = 𝑧)
158, 14sylan2b 275 . . . 4 ((∀𝑦(𝑦𝑥𝜑) ∧ [𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑)) → 𝑥 = 𝑧)
1615gen2 1355 . . 3 𝑥𝑧((∀𝑦(𝑦𝑥𝜑) ∧ [𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑)) → 𝑥 = 𝑧)
1716jctr 302 . 2 (∃𝑥𝑦(𝑦𝑥𝜑) → (∃𝑥𝑦(𝑦𝑥𝜑) ∧ ∀𝑥𝑧((∀𝑦(𝑦𝑥𝜑) ∧ [𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑)) → 𝑥 = 𝑧)))
18 nfv 1437 . . 3 𝑧𝑦(𝑦𝑥𝜑)
1918eu2 1960 . 2 (∃!𝑥𝑦(𝑦𝑥𝜑) ↔ (∃𝑥𝑦(𝑦𝑥𝜑) ∧ ∀𝑥𝑧((∀𝑦(𝑦𝑥𝜑) ∧ [𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑)) → 𝑥 = 𝑧)))
2017, 19sylibr 141 1 (∃𝑥𝑦(𝑦𝑥𝜑) → ∃!𝑥𝑦(𝑦𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  wnf 1365  wex 1397  [wsb 1661  ∃!weu 1916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919
This theorem is referenced by:  zfnuleu  3908
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