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Theorem mobid 2054
Description: Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.)
Hypotheses
Ref Expression
mobid.1 𝑥𝜑
mobid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
mobid (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Proof of Theorem mobid
StepHypRef Expression
1 mobid.1 . . . 4 𝑥𝜑
2 mobid.2 . . . 4 (𝜑 → (𝜓𝜒))
31, 2exbid 1609 . . 3 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
41, 2eubid 2026 . . 3 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
53, 4imbi12d 233 . 2 (𝜑 → ((∃𝑥𝜓 → ∃!𝑥𝜓) ↔ (∃𝑥𝜒 → ∃!𝑥𝜒)))
6 df-mo 2023 . 2 (∃*𝑥𝜓 ↔ (∃𝑥𝜓 → ∃!𝑥𝜓))
7 df-mo 2023 . 2 (∃*𝑥𝜒 ↔ (∃𝑥𝜒 → ∃!𝑥𝜒))
85, 6, 73bitr4g 222 1 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wnf 1453  wex 1485  ∃!weu 2019  ∃*wmo 2020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-eu 2022  df-mo 2023
This theorem is referenced by:  mobidv  2055  rmobida  2656  rmoeq1f  2664
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