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Theorem mobidh 1977
Description: Formula-building rule for "at most one" quantifier (deduction rule). (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 1546 . . 3 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
41, 2eubidh 1949 . . 3 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
53, 4imbi12d 232 . 2 (𝜑 → ((∃𝑥𝜓 → ∃!𝑥𝜓) ↔ (∃𝑥𝜒 → ∃!𝑥𝜒)))
6 df-mo 1947 . 2 (∃*𝑥𝜓 ↔ (∃𝑥𝜓 → ∃!𝑥𝜓))
7 df-mo 1947 . 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 1283  wex 1422  ∃!weu 1943  ∃*wmo 1944
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-eu 1946  df-mo 1947
This theorem is referenced by:  euan  1999
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