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Theorem sb8mo 2603
Description: Variable substitution for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Alexander van der Vekens, 17-Jun-2017.) (New usage is discouraged.)
Hypothesis
Ref Expression
sb8eu.1 𝑦𝜑
Assertion
Ref Expression
sb8mo (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8mo
StepHypRef Expression
1 sb8eu.1 . . . 4 𝑦𝜑
21sb8e 2524 . . 3 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
31sb8eu 2602 . . 3 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
42, 3imbi12i 351 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
5 moeu 2585 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
6 moeu 2585 . 2 (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
74, 5, 63bitr4i 303 1 (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wex 1786  wnf 1790  [wsb 2071  ∃*wmo 2540  ∃!weu 2570
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 2015  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2374
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571
This theorem is referenced by:  cbvmo  2607
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