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Theorem sbmo 2544
 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 2491 . . 3 ([𝑦 / 𝑥]∃𝑤𝑧(𝜑𝑧 = 𝑤) ↔ ∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑𝑧 = 𝑤))
2 nfv 1883 . . . . . 6 𝑥 𝑧 = 𝑤
32sblim 2425 . . . . 5 ([𝑦 / 𝑥](𝜑𝑧 = 𝑤) ↔ ([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
43sbalv 2492 . . . 4 ([𝑦 / 𝑥]∀𝑧(𝜑𝑧 = 𝑤) ↔ ∀𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
54exbii 1814 . . 3 (∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑𝑧 = 𝑤) ↔ ∃𝑤𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
61, 5bitri 264 . 2 ([𝑦 / 𝑥]∃𝑤𝑧(𝜑𝑧 = 𝑤) ↔ ∃𝑤𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
7 mo2v 2505 . . 3 (∃*𝑧𝜑 ↔ ∃𝑤𝑧(𝜑𝑧 = 𝑤))
87sbbii 1944 . 2 ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ [𝑦 / 𝑥]∃𝑤𝑧(𝜑𝑧 = 𝑤))
9 mo2v 2505 . 2 (∃*𝑧[𝑦 / 𝑥]𝜑 ↔ ∃𝑤𝑧([𝑦 / 𝑥]𝜑𝑧 = 𝑤))
106, 8, 93bitr4i 292 1 ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1521  ∃wex 1744  [wsb 1937  ∃*wmo 2499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503 This theorem is referenced by: (None)
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