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Theorem mo3 1999
Description: Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that 𝑦 not occur in 𝜑 in place of our hypothesis. (Contributed by NM, 8-Mar-1995.)
Hypothesis
Ref Expression
mo3.1 𝑦𝜑
Assertion
Ref Expression
mo3 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mo3
StepHypRef Expression
1 mo3.1 . . 3 𝑦𝜑
21nfri 1455 . 2 (𝜑 → ∀𝑦𝜑)
32mo3h 1998 1 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1285  wnf 1392  [wsb 1689  ∃*wmo 1946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949
This theorem is referenced by:  sbmo  2004  rmo3  2918  isarep2  5057
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