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Theorem mobid 1951
Description: Formula-building rule for "at most one" quantifier (deduction rule). (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 1523 . . 3 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
41, 2eubid 1923 . . 3 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
53, 4imbi12d 227 . 2 (𝜑 → ((∃𝑥𝜓 → ∃!𝑥𝜓) ↔ (∃𝑥𝜒 → ∃!𝑥𝜒)))
6 df-mo 1920 . 2 (∃*𝑥𝜓 ↔ (∃𝑥𝜓 → ∃!𝑥𝜓))
7 df-mo 1920 . 2 (∃*𝑥𝜒 ↔ (∃𝑥𝜒 → ∃!𝑥𝜒))
85, 6, 73bitr4g 216 1 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wnf 1365  wex 1397  ∃!weu 1916  ∃*wmo 1917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-eu 1919  df-mo 1920
This theorem is referenced by:  mobidv  1952  rmobida  2513  rmoeq1f  2521
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