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Theorem sbmo 2498
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 sbex 2446 . . 3 ([𝑦 / 𝑥]∃𝑤𝑧(𝜑𝑧 = 𝑤) ↔ ∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑𝑧 = 𝑤))
2 nfv 1828 . . . . . 6 𝑥 𝑧 = 𝑤
32sblim 2380 . . . . 5 ([𝑦 / 𝑥](𝜑𝑧 = 𝑤) ↔ ([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
43sbalv 2447 . . . 4 ([𝑦 / 𝑥]∀𝑧(𝜑𝑧 = 𝑤) ↔ ∀𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
54exbii 1762 . . 3 (∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑𝑧 = 𝑤) ↔ ∃𝑤𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
61, 5bitri 262 . 2 ([𝑦 / 𝑥]∃𝑤𝑧(𝜑𝑧 = 𝑤) ↔ ∃𝑤𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
7 mo2v 2460 . . 3 (∃*𝑧𝜑 ↔ ∃𝑤𝑧(𝜑𝑧 = 𝑤))
87sbbii 1872 . 2 ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ [𝑦 / 𝑥]∃𝑤𝑧(𝜑𝑧 = 𝑤))
9 mo2v 2460 . 2 (∃*𝑧[𝑦 / 𝑥]𝜑 ↔ ∃𝑤𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
106, 8, 93bitr4i 290 1 ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  wex 1694  [wsb 1865  ∃*wmo 2454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458
This theorem is referenced by: (None)
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