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Theorem moi2 2716
 Description: Consequence of "at most one." (Contributed by NM, 29-Jun-2008.)
Hypothesis
Ref Expression
moi2.1 (x = A → (φψ))
Assertion
Ref Expression
moi2 (((A B ∃*xφ) (φ ψ)) → x = A)
Distinct variable groups:   x,A   ψ,x
Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem moi2
StepHypRef Expression
1 moi2.1 . . . . 5 (x = A → (φψ))
21mob2 2715 . . . 4 ((A B ∃*xφ φ) → (x = Aψ))
323expa 1103 . . 3 (((A B ∃*xφ) φ) → (x = Aψ))
43biimprd 147 . 2 (((A B ∃*xφ) φ) → (ψx = A))
54impr 361 1 (((A B ∃*xφ) (φ ψ)) → x = A)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = 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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