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Theorem mobidh 1982
 Description: Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.)
Hypotheses
Ref Expression
mobidh.1 (𝜑 → ∀𝑥𝜑)
mobidh.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
mobidh (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Proof of Theorem mobidh
StepHypRef Expression
1 mobidh.1 . . . 4 (𝜑 → ∀𝑥𝜑)
2 mobidh.2 . . . 4 (𝜑 → (𝜓𝜒))
31, 2exbidh 1550 . . 3 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
41, 2eubidh 1954 . . 3 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
53, 4imbi12d 232 . 2 (𝜑 → ((∃𝑥𝜓 → ∃!𝑥𝜓) ↔ (∃𝑥𝜒 → ∃!𝑥𝜒)))
6 df-mo 1952 . 2 (∃*𝑥𝜓 ↔ (∃𝑥𝜓 → ∃!𝑥𝜓))
7 df-mo 1952 . 2 (∃*𝑥𝜒 ↔ (∃𝑥𝜒 → ∃!𝑥𝜒))
85, 6, 73bitr4g 221 1 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1287  ∃wex 1426  ∃!weu 1948  ∃*wmo 1949 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472 This theorem depends on definitions:  df-bi 115  df-eu 1951  df-mo 1952 This theorem is referenced by:  euan  2004
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