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Theorem mo4 2016
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 1473 . 2 𝑥𝜓
2 mo4.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2mo4f 2015 1 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1294  ∃*wmo 1956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480
This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959
This theorem is referenced by:  eu4  2017  rmo4  2822  dffun5r  5061  dffun6f  5062  fun11  5115  brprcneu  5333  dff13  5585  mpt2fun  5785  caovimo  5876  th3qlem1  6434  addnq0mo  7103  mulnq0mo  7104  addsrmo  7386  mulsrmo  7387  summodc  10941
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