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Theorem mobid 2550
Description: Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) Remove dependency on ax-10 2142, ax-11 2159, ax-13 2372. (Revised by BJ, 14-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.)
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 . . 3 𝑥𝜑
2 mobid.2 . . 3 (𝜑 → (𝜓𝜒))
31, 2alrimi 2212 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 mobi 2547 . 2 (∀𝑥(𝜓𝜒) → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
53, 4syl 17 1 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1541  wnf 1791  ∃*wmo 2538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-nf 1792  df-mo 2540
This theorem is referenced by:  moanim  2622  rmobida  3316  rmoeq1f  3324  funcnvmpt  30748
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