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Theorem mo2 2507
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 2505 . 2 (∃*𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 mo2.1 . . . . 5 𝑦𝜑
3 nfv 1883 . . . . 5 𝑦 𝑥 = 𝑧
42, 3nfim 1865 . . . 4 𝑦(𝜑𝑥 = 𝑧)
54nfal 2191 . . 3 𝑦𝑥(𝜑𝑥 = 𝑧)
6 nfv 1883 . . 3 𝑧𝑥(𝜑𝑥 = 𝑦)
7 equequ2 1999 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
87imbi2d 329 . . . 4 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜑𝑥 = 𝑦)))
98albidv 1889 . . 3 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
105, 6, 9cbvex 2308 . 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 1521  wex 1744  wnf 1748  ∃*wmo 2499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-eu 2502  df-mo 2503
This theorem is referenced by:  mo3  2536  mo  2537  rmo2  3559  nmo  29453  bj-eu3f  32954  bj-mo3OLD  32957  dffun3f  42754
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