Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbmo Structured version   Visualization version   GIF version

Theorem sbmo 2697
 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 2288 . . 3 ([𝑦 / 𝑥]∃𝑤𝑧(𝜑𝑧 = 𝑤) ↔ ∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑𝑧 = 𝑤))
2 nfv 1915 . . . . . 6 𝑥 𝑧 = 𝑤
32sblim 2315 . . . . 5 ([𝑦 / 𝑥](𝜑𝑧 = 𝑤) ↔ ([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
43sbalv 2167 . . . 4 ([𝑦 / 𝑥]∀𝑧(𝜑𝑧 = 𝑤) ↔ ∀𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
54exbii 1848 . . 3 (∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑𝑧 = 𝑤) ↔ ∃𝑤𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
61, 5bitri 277 . 2 ([𝑦 / 𝑥]∃𝑤𝑧(𝜑𝑧 = 𝑤) ↔ ∃𝑤𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
7 df-mo 2622 . . 3 (∃*𝑧𝜑 ↔ ∃𝑤𝑧(𝜑𝑧 = 𝑤))
87sbbii 2081 . 2 ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ [𝑦 / 𝑥]∃𝑤𝑧(𝜑𝑧 = 𝑤))
9 df-mo 2622 . 2 (∃*𝑧[𝑦 / 𝑥]𝜑 ↔ ∃𝑤𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
106, 8, 93bitr4i 305 1 ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535  ∃wex 1780  [wsb 2069  ∃*wmo 2620 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator