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Theorem moi2 3017
 Description: Consequence of "at most one." (Contributed by NM, 29-Jun-2008.)
Hypothesis
Ref Expression
moi2.1 (x = A → (φψ))
Assertion
Ref Expression
moi2 (((A B ∃*xφ) (φ ψ)) → 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 3016 . . . 4 ((A B ∃*xφ φ) → (x = Aψ))
323expa 1151 . . 3 (((A B ∃*xφ) φ) → (x = Aψ))
43biimprd 214 . 2 (((A B ∃*xφ) φ) → (ψx = A))
54impr 602 1 (((A B ∃*xφ) (φ ψ)) → x = A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861 This theorem is referenced by: (None)
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