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Theorem sbmo 2618
Description: Substitution into an at-most-one quantifier. (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 sbex 2286 . . 3 ([𝑦 / 𝑥]∃𝑤𝑧(𝜑𝑧 = 𝑤) ↔ ∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑𝑧 = 𝑤))
2 nfv 1921 . . . . . 6 𝑥 𝑧 = 𝑤
32sblim 2311 . . . . 5 ([𝑦 / 𝑥](𝜑𝑧 = 𝑤) ↔ ([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
43sbalv 2168 . . . 4 ([𝑦 / 𝑥]∀𝑧(𝜑𝑧 = 𝑤) ↔ ∀𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
54exbii 1854 . . 3 (∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑𝑧 = 𝑤) ↔ ∃𝑤𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
61, 5bitri 278 . 2 ([𝑦 / 𝑥]∃𝑤𝑧(𝜑𝑧 = 𝑤) ↔ ∃𝑤𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
7 df-mo 2541 . . 3 (∃*𝑧𝜑 ↔ ∃𝑤𝑧(𝜑𝑧 = 𝑤))
87sbbii 2086 . 2 ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ [𝑦 / 𝑥]∃𝑤𝑧(𝜑𝑧 = 𝑤))
9 df-mo 2541 . 2 (∃*𝑧[𝑦 / 𝑥]𝜑 ↔ ∃𝑤𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
106, 8, 93bitr4i 306 1 ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1540  wex 1786  [wsb 2074  ∃*wmo 2539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-11 2162  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541
This theorem is referenced by: (None)
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