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Theorem moi 3019
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 (((A C B D) ∃*xφ (ψ χ)) → 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 3018 . . . . 5 (((A C B D) ∃*xφ ψ) → (A = Bχ))
43biimprd 214 . . . 4 (((A C B D) ∃*xφ ψ) → (χA = B))
543expia 1153 . . 3 (((A C B D) ∃*xφ) → (ψ → (χA = B)))
65imp3a 420 . 2 (((A C B D) ∃*xφ) → ((ψ χ) → A = B))
763impia 1148 1 (((A C B D) ∃*xφ (ψ χ)) → A = B)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176   wa 358   w3a 934   = wceq 1642   wcel 1710  ∃*wmo 2205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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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