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Theorem sbmo 2085
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 1528 . . . . . 6 𝑥 𝑧 = 𝑤
21sblim 1957 . . . . 5 ([𝑦 / 𝑥]((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ ([𝑦 / 𝑥](𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
3 sban 1955 . . . . . 6 ([𝑦 / 𝑥](𝜑 ∧ [𝑤 / 𝑧]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
43imbi1i 238 . . . . 5 (([𝑦 / 𝑥](𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
5 sbcom2 1987 . . . . . . 7 ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑)
65anbi2i 457 . . . . . 6 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑))
76imbi1i 238 . . . . 5 ((([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ (([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
82, 4, 73bitri 206 . . . 4 ([𝑦 / 𝑥]((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ (([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
98sbalv 2005 . . 3 ([𝑦 / 𝑥]∀𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ ∀𝑤(([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
109sbalv 2005 . 2 ([𝑦 / 𝑥]∀𝑧𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ ∀𝑧𝑤(([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
11 nfv 1528 . . . 4 𝑤𝜑
1211mo3 2080 . . 3 (∃*𝑧𝜑 ↔ ∀𝑧𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
1312sbbii 1765 . 2 ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ [𝑦 / 𝑥]∀𝑧𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
14 nfv 1528 . . 3 𝑤[𝑦 / 𝑥]𝜑
1514mo3 2080 . 2 (∃*𝑧[𝑦 / 𝑥]𝜑 ↔ ∀𝑧𝑤(([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
1610, 13, 153bitr4i 212 1 ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351   = wceq 1353  [wsb 1762  ∃*wmo 2027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030
This theorem is referenced by: (None)
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