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Theorem moi2 3704
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 3703 . . . 4 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))
323expa 1110 . . 3 (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ 𝜑) → (𝑥 = 𝐴𝜓))
43biimprd 249 . 2 (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ 𝜑) → (𝜓𝑥 = 𝐴))
54impr 455 1 (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑𝜓)) → 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  ∃*wmo 2613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-clab 2797  df-cleq 2811  df-clel 2890  df-v 3494
This theorem is referenced by:  fsum  15065  fprod  15283  txcn  22162  haustsms2  22672
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