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Theorem mo4f 2060
 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 ax-17 1507 . . 3 (𝜑 → ∀𝑦𝜑)
21mo3h 2053 . 2 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
3 mo4f.1 . . . . . 6 𝑥𝜓
4 mo4f.2 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbie 1765 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
65anbi2i 453 . . . 4 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑𝜓))
76imbi1i 237 . . 3 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑𝜓) → 𝑥 = 𝑦))
872albii 1448 . 2 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
92, 8bitri 183 1 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330  Ⅎwnf 1437  [wsb 1736  ∃*wmo 2001 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004 This theorem is referenced by:  mo4  2061  mob2  2869  moop2  4182  dffun4f  5148
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