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Theorem moi2 3710
Description: Consequence of "at most one". (Contributed by NM, 29-Jun-2008.)
Hypothesis
Ref Expression
moi2.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
moi2 (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑𝜓)) → 𝑥 = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem moi2
StepHypRef Expression
1 moi2.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
21mob2 3709 . . . 4 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))
323expa 1116 . . 3 (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ 𝜑) → (𝑥 = 𝐴𝜓))
43biimprd 247 . 2 (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ 𝜑) → (𝜓𝑥 = 𝐴))
54impr 454 1 (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑𝜓)) → 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  ∃*wmo 2528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472
This theorem is referenced by:  fsum  15693  fprod  15912  txcn  23524  haustsms2  24035
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