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Theorem mobid 2618
 Description: Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) Remove dependency on ax-10 2192, ax-11 2207, ax-13 2389. (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 2256 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 mobi 2613 . 2 (∀𝑥(𝜓𝜒) → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
53, 4syl 17 1 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1654  Ⅎwnf 1882  ∃*wmo 2603 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-12 2220 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-nf 1883  df-mo 2605 This theorem is referenced by:  eubidOLD  2661  mobidvOLD  2679  moanim  2708  rmobida  3341  rmoeq1f  3349  funcnvmpt  30012
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