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Theorem mo4 2087
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 1528 . 2 𝑥𝜓
2 mo4.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2mo4f 2086 1 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351  ∃*wmo 2027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030
This theorem is referenced by:  eu4  2088  rmo4  2932  dffun5r  5230  dffun6f  5231  fun11  5285  brprcneu  5510  dff13  5772  mpofun  5980  caovimo  6071  th3qlem1  6640  exmidmotap  7263  addnq0mo  7449  mulnq0mo  7450  addsrmo  7745  mulsrmo  7746  summodc  11394  prodmodc  11589  limcimo  14295
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