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Theorem mo2icl 2697
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 1416 . . . . 5 xx(φx = A)
2 vex 2538 . . . . . . . 8 x V
3 eleq1 2082 . . . . . . . 8 (x = A → (x V ↔ A V))
42, 3mpbii 136 . . . . . . 7 (x = AA V)
54imim2i 12 . . . . . 6 ((φx = A) → (φA V))
65sps 1412 . . . . 5 (x(φx = A) → (φA V))
71, 6eximd 1485 . . . 4 (x(φx = A) → (xφx A V))
8 19.9v 1733 . . . 4 (x A V ↔ A V)
97, 8syl6ib 150 . . 3 (x(φx = A) → (xφA V))
10 eqeq2 2031 . . . . . . . 8 (y = A → (x = yx = A))
1110imbi2d 219 . . . . . . 7 (y = A → ((φx = y) ↔ (φx = A)))
1211albidv 1687 . . . . . 6 (y = A → (x(φx = y) ↔ x(φx = A)))
1312imbi1d 220 . . . . 5 (y = A → ((x(φx = y) → ∃*xφ) ↔ (x(φx = A) → ∃*xφ)))
14 nfv 1402 . . . . . . 7 yφ
1514mo2r 1934 . . . . . 6 (yx(φx = y) → ∃*xφ)
161519.23bi 1465 . . . . 5 (x(φx = y) → ∃*xφ)
1713, 16vtoclg 2590 . . . 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 1931 . 2 (∃*xφ ↔ (xφ∃*xφ))
2119, 20sylibr 137 1 (x(φx = A) → ∃*xφ)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1226   = wceq 1228  wex 1362   wcel 1374  ∃*wmo 1883  Vcvv 2535
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537
This theorem is referenced by:  invdisj  3733
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