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Theorem moi 2718
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 2717 . . . . 5 (((A 𝐶 B 𝐷) ∃*xφ ψ) → (A = Bχ))
43biimprd 147 . . . 4 (((A 𝐶 B 𝐷) ∃*xφ ψ) → (χA = B))
543expia 1105 . . 3 (((A 𝐶 B 𝐷) ∃*xφ) → (ψ → (χA = B)))
65impd 242 . 2 (((A 𝐶 B 𝐷) ∃*xφ) → ((ψ χ) → A = B))
763impia 1100 1 (((A 𝐶 B 𝐷) ∃*xφ (ψ χ)) → A = B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  ∃*wmo 1898
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553
This theorem is referenced by: (None)
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