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Theorem mo4 1961
Description: "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)
Hypothesis
Ref Expression
mo4.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
mo4 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem mo4
StepHypRef Expression
1 nfv 1421 . 2 𝑥𝜓
2 mo4.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2mo4f 1960 1 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wal 1241  ∃*wmo 1901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904
This theorem is referenced by:  eu4  1962  rmo4  2734  dffun5r  4914  dffun6f  4915  fun11  4966  brprcneu  5171  dff13  5407  mpt2fun  5603  caovimo  5694  th3qlem1  6208  addnq0mo  6545  mulnq0mo  6546  addsrmo  6828  mulsrmo  6829
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