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Theorem sb8mo 2503
Description: Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
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 2424 . . 3 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
31sb8eu 2502 . . 3 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
42, 3imbi12i 340 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
5 df-mo 2474 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
6 df-mo 2474 . 2 (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
74, 5, 63bitr4i 292 1 (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wex 1701  wnf 1705  [wsb 1877  ∃!weu 2469  ∃*wmo 2470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474
This theorem is referenced by: (None)
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