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Theorem moi 2724
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 2723 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))
43biimprd 147 . . . 4 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝜒𝐴 = 𝐵))
543expia 1106 . . 3 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑) → (𝜓 → (𝜒𝐴 = 𝐵)))
65impd 242 . 2 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑) → ((𝜓𝜒) → 𝐴 = 𝐵))
763impia 1101 1 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑 ∧ (𝜓𝜒)) → 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wcel 1393  ∃*wmo 1901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559
This theorem is referenced by: (None)
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