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Theorem mo2icl 2792
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 1479 . . . . 5 𝑥𝑥(𝜑𝑥 = 𝐴)
2 vex 2622 . . . . . . . 8 𝑥 ∈ V
3 eleq1 2150 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
42, 3mpbii 146 . . . . . . 7 (𝑥 = 𝐴𝐴 ∈ V)
54imim2i 12 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝜑𝐴 ∈ V))
65sps 1475 . . . . 5 (∀𝑥(𝜑𝑥 = 𝐴) → (𝜑𝐴 ∈ V))
71, 6eximd 1548 . . . 4 (∀𝑥(𝜑𝑥 = 𝐴) → (∃𝑥𝜑 → ∃𝑥 𝐴 ∈ V))
8 19.9v 1799 . . . 4 (∃𝑥 𝐴 ∈ V ↔ 𝐴 ∈ V)
97, 8syl6ib 159 . . 3 (∀𝑥(𝜑𝑥 = 𝐴) → (∃𝑥𝜑𝐴 ∈ V))
10 eqeq2 2097 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
1110imbi2d 228 . . . . . . 7 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
1211albidv 1752 . . . . . 6 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
1312imbi1d 229 . . . . 5 (𝑦 = 𝐴 → ((∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)))
14 nfv 1466 . . . . . . 7 𝑦𝜑
1514mo2r 2000 . . . . . 6 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
161519.23bi 1528 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
1713, 16vtoclg 2679 . . . 4 (𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑))
1817com12 30 . . 3 (∀𝑥(𝜑𝑥 = 𝐴) → (𝐴 ∈ V → ∃*𝑥𝜑))
199, 18syld 44 . 2 (∀𝑥(𝜑𝑥 = 𝐴) → (∃𝑥𝜑 → ∃*𝑥𝜑))
20 moabs 1997 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
2119, 20sylibr 132 1 (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287   = wceq 1289  wex 1426  wcel 1438  ∃*wmo 1949  Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621
This theorem is referenced by:  invdisj  3831
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