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Theorem mo4f 2514
Description: "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
Hypotheses
Ref Expression
mo4f.1 𝑥𝜓
mo4f.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
mo4f (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem mo4f
StepHypRef Expression
1 nfv 1841 . . 3 𝑦𝜑
21mo3 2505 . 2 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
3 mo4f.1 . . . . . 6 𝑥𝜓
4 mo4f.2 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbie 2406 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
65anbi2i 729 . . . 4 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑𝜓))
76imbi1i 339 . . 3 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑𝜓) → 𝑥 = 𝑦))
872albii 1746 . 2 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
92, 8bitri 264 1 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1479  wnf 1706  [wsb 1878  ∃*wmo 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473
This theorem is referenced by:  mo4  2515  bm1.1  2605  mob2  3380  moop2  4956
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