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Theorem mo2icl 2714
 Description: Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)
Assertion
Ref Expression
mo2icl (x(φx = A) → ∃*xφ)
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem mo2icl
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 nfa1 1431 . . . . 5 xx(φx = A)
2 vex 2554 . . . . . . . 8 x V
3 eleq1 2097 . . . . . . . 8 (x = A → (x V ↔ A V))
42, 3mpbii 136 . . . . . . 7 (x = AA V)
54imim2i 12 . . . . . 6 ((φx = A) → (φA V))
65sps 1427 . . . . 5 (x(φx = A) → (φA V))
71, 6eximd 1500 . . . 4 (x(φx = A) → (xφx A V))
8 19.9v 1748 . . . 4 (x A V ↔ A V)
97, 8syl6ib 150 . . 3 (x(φx = A) → (xφA V))
10 eqeq2 2046 . . . . . . . 8 (y = A → (x = yx = A))
1110imbi2d 219 . . . . . . 7 (y = A → ((φx = y) ↔ (φx = A)))
1211albidv 1702 . . . . . 6 (y = A → (x(φx = y) ↔ x(φx = A)))
1312imbi1d 220 . . . . 5 (y = A → ((x(φx = y) → ∃*xφ) ↔ (x(φx = A) → ∃*xφ)))
14 nfv 1418 . . . . . . 7 yφ
1514mo2r 1949 . . . . . 6 (yx(φx = y) → ∃*xφ)
161519.23bi 1480 . . . . 5 (x(φx = y) → ∃*xφ)
1713, 16vtoclg 2607 . . . 4 (A V → (x(φx = A) → ∃*xφ))
1817com12 27 . . 3 (x(φx = A) → (A V → ∃*xφ))
199, 18syld 40 . 2 (x(φx = A) → (xφ∃*xφ))
20 moabs 1946 . 2 (∃*xφ ↔ (xφ∃*xφ))
2119, 20sylibr 137 1 (x(φx = A) → ∃*xφ)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃*wmo 1898  Vcvv 2551 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-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:  invdisj  3750
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