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Theorem moi 3706
Description: Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
Hypotheses
Ref Expression
moi.1 (𝑥 = 𝐴 → (𝜑𝜓))
moi.2 (𝑥 = 𝐵 → (𝜑𝜒))
Assertion
Ref Expression
moi (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓𝜒)) → 𝐴 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜒,𝑥   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem moi
StepHypRef Expression
1 moi.1 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
2 moi.2 . . . . . 6 (𝑥 = 𝐵 → (𝜑𝜒))
31, 2mob 3705 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))
43biimprd 249 . . . 4 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝜒𝐴 = 𝐵))
543expia 1113 . . 3 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑) → (𝜓 → (𝜒𝐴 = 𝐵)))
65impd 411 . 2 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑) → ((𝜓𝜒) → 𝐴 = 𝐵))
763impia 1109 1 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓𝜒)) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = 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:  enqeq  10344  f1otrspeq  18504  hausflim  22517  tglineineq  26356  tglineinteq  26358
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