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Theorem moi 3674
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 3673 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))
43biimprd 247 . . . 4 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝜒𝐴 = 𝐵))
543expia 1121 . . 3 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑) → (𝜓 → (𝜒𝐴 = 𝐵)))
65impd 411 . 2 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑) → ((𝜓𝜒) → 𝐴 = 𝐵))
763impia 1117 1 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓𝜒)) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  ∃*wmo 2536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3445
This theorem is referenced by:  enqeq  10828  f1otrspeq  19182  hausflim  23278  tglineineq  27430  tglineinteq  27432
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