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Theorem mo2 2483
Description: Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.) Restrict dummy variable z. (Revised by Wolf Lammen, 28-May-2019.)
Hypothesis
Ref Expression
mo2.1 𝑦𝜑
Assertion
Ref Expression
mo2 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mo2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 mo2v 2481 . 2 (∃*𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 mo2.1 . . . . 5 𝑦𝜑
3 nfv 1845 . . . . 5 𝑦 𝑥 = 𝑧
42, 3nfim 1827 . . . 4 𝑦(𝜑𝑥 = 𝑧)
54nfal 2155 . . 3 𝑦𝑥(𝜑𝑥 = 𝑧)
6 nfv 1845 . . 3 𝑧𝑥(𝜑𝑥 = 𝑦)
7 equequ2 1955 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
87imbi2d 330 . . . 4 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜑𝑥 = 𝑦)))
98albidv 1851 . . 3 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
105, 6, 9cbvex 2276 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
111, 10bitri 264 1 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wex 1701  wnf 1705  ∃*wmo 2475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-eu 2478  df-mo 2479
This theorem is referenced by:  mo3  2511  mo  2512  rmo2  3512  nmo  29165  bj-eu3f  32464  bj-mo3OLD  32469  dffun3f  41695
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