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Theorem moi 3727
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 3726 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))
43biimprd 248 . . . 4 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝜒𝐴 = 𝐵))
543expia 1120 . . 3 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑) → (𝜓 → (𝜒𝐴 = 𝐵)))
65impd 410 . 2 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑) → ((𝜓𝜒) → 𝐴 = 𝐵))
763impia 1116 1 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓𝜒)) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  ∃*wmo 2536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480
This theorem is referenced by:  enqeq  10972  f1otrspeq  19480  hausflim  24005  tglineineq  28666  tglineinteq  28668
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