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Theorem moi2 3613
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 3612 . . . 4 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))
323expa 1153 . . 3 (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ 𝜑) → (𝑥 = 𝐴𝜓))
43biimprd 240 . 2 (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ 𝜑) → (𝜓𝑥 = 𝐴))
54impr 448 1 (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑𝜓)) → 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  ∃*wmo 2604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2804
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-v 3417
This theorem is referenced by:  fsum  14829  fprod  15045  txcn  21801  haustsms2  22311
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