ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  bm1.1 GIF version

Theorem bm1.1 2142
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 1569 . . . . . . 7 𝑥(𝑦𝑧𝜑)
43nfal 1556 . . . . . 6 𝑥𝑦(𝑦𝑧𝜑)
5 elequ2 2133 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
65bibi1d 232 . . . . . . 7 (𝑥 = 𝑧 → ((𝑦𝑥𝜑) ↔ (𝑦𝑧𝜑)))
76albidv 1804 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦(𝑦𝑥𝜑) ↔ ∀𝑦(𝑦𝑧𝜑)))
84, 7sbie 1771 . . . . 5 ([𝑧 / 𝑥]∀𝑦(𝑦𝑥𝜑) ↔ ∀𝑦(𝑦𝑧𝜑))
9 19.26 1461 . . . . . 6 (∀𝑦((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) ↔ (∀𝑦(𝑦𝑥𝜑) ∧ ∀𝑦(𝑦𝑧𝜑)))
10 biantr 937 . . . . . . . 8 (((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) → (𝑦𝑥𝑦𝑧))
1110alimi 1435 . . . . . . 7 (∀𝑦((𝑦𝑥𝜑) ∧ (𝑦𝑧𝜑)) → ∀𝑦(𝑦𝑥𝑦𝑧))
12 ax-ext 2139 . . . . . . 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 1430 . . 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 2050 . 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 1333  wnf 1440  wex 1472  [wsb 1742  ∃!weu 2006
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009
This theorem is referenced by:  zfnuleu  4090
  Copyright terms: Public domain W3C validator