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Theorem sbmo 2001
Description: Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
sbmo ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbmo
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 1462 . . . . . 6 𝑥 𝑧 = 𝑤
21sblim 1873 . . . . 5 ([𝑦 / 𝑥]((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ ([𝑦 / 𝑥](𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
3 sban 1871 . . . . . 6 ([𝑦 / 𝑥](𝜑 ∧ [𝑤 / 𝑧]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
43imbi1i 236 . . . . 5 (([𝑦 / 𝑥](𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
5 sbcom2 1905 . . . . . . 7 ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑)
65anbi2i 445 . . . . . 6 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑))
76imbi1i 236 . . . . 5 ((([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ (([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
82, 4, 73bitri 204 . . . 4 ([𝑦 / 𝑥]((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ (([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
98sbalv 1923 . . 3 ([𝑦 / 𝑥]∀𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ ∀𝑤(([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
109sbalv 1923 . 2 ([𝑦 / 𝑥]∀𝑧𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ ∀𝑧𝑤(([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
11 nfv 1462 . . . 4 𝑤𝜑
1211mo3 1996 . . 3 (∃*𝑧𝜑 ↔ ∀𝑧𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
1312sbbii 1689 . 2 ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ [𝑦 / 𝑥]∀𝑧𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
14 nfv 1462 . . 3 𝑤[𝑦 / 𝑥]𝜑
1514mo3 1996 . 2 (∃*𝑧[𝑦 / 𝑥]𝜑 ↔ ∀𝑧𝑤(([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
1610, 13, 153bitr4i 210 1 ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  [wsb 1686  ∃*wmo 1943
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946
This theorem is referenced by: (None)
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