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Theorem sbmo 2059
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 1509 . . . . . 6 𝑥 𝑧 = 𝑤
21sblim 1931 . . . . 5 ([𝑦 / 𝑥]((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ ([𝑦 / 𝑥](𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
3 sban 1929 . . . . . 6 ([𝑦 / 𝑥](𝜑 ∧ [𝑤 / 𝑧]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
43imbi1i 237 . . . . 5 (([𝑦 / 𝑥](𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
5 sbcom2 1963 . . . . . . 7 ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑)
65anbi2i 453 . . . . . 6 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑))
76imbi1i 237 . . . . 5 ((([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ (([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
82, 4, 73bitri 205 . . . 4 ([𝑦 / 𝑥]((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ (([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
98sbalv 1981 . . 3 ([𝑦 / 𝑥]∀𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ ∀𝑤(([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
109sbalv 1981 . 2 ([𝑦 / 𝑥]∀𝑧𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ ∀𝑧𝑤(([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
11 nfv 1509 . . . 4 𝑤𝜑
1211mo3 2054 . . 3 (∃*𝑧𝜑 ↔ ∀𝑧𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
1312sbbii 1739 . 2 ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ [𝑦 / 𝑥]∀𝑧𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
14 nfv 1509 . . 3 𝑤[𝑦 / 𝑥]𝜑
1514mo3 2054 . 2 (∃*𝑧[𝑦 / 𝑥]𝜑 ↔ ∀𝑧𝑤(([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
1610, 13, 153bitr4i 211 1 ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1330   = wceq 1332  [wsb 1736  ∃*wmo 2001
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004
This theorem is referenced by: (None)
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