MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mobid Structured version   Visualization version   GIF version

Theorem mobid 2632
Description: Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) Remove dependency on ax-10 2138, ax-11 2153, ax-13 2385. (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 2206 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 mobi 2628 . 2 (∀𝑥(𝜓𝜒) → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
53, 4syl 17 1 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1528  wnf 1777  ∃*wmo 2618
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 1904  ax-6 1963  ax-7 2008  ax-12 2169
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-nf 1778  df-mo 2620
This theorem is referenced by:  moanim  2704  rmobida  3398  rmoeq1f  3406  funcnvmpt  30346
  Copyright terms: Public domain W3C validator