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Theorem mobid 2536
Description: Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) Remove dependency on ax-10 2129, ax-11 2146, ax-13 2363. (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 2198 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 mobi 2533 . 2 (∀𝑥(𝜓𝜒) → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
53, 4syl 17 1 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wnf 1777  ∃*wmo 2524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778  df-mo 2526
This theorem is referenced by:  moanim  2608  rmobida  3394  rmoeq1f  3412  funcnvmpt  32386
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