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Theorem mo2icl 2863
 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 1521 . . . . 5 𝑥𝑥(𝜑𝑥 = 𝐴)
2 vex 2689 . . . . . . . 8 𝑥 ∈ V
3 eleq1 2202 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
42, 3mpbii 147 . . . . . . 7 (𝑥 = 𝐴𝐴 ∈ V)
54imim2i 12 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝜑𝐴 ∈ V))
65sps 1517 . . . . 5 (∀𝑥(𝜑𝑥 = 𝐴) → (𝜑𝐴 ∈ V))
71, 6eximd 1591 . . . 4 (∀𝑥(𝜑𝑥 = 𝐴) → (∃𝑥𝜑 → ∃𝑥 𝐴 ∈ V))
8 19.9v 1843 . . . 4 (∃𝑥 𝐴 ∈ V ↔ 𝐴 ∈ V)
97, 8syl6ib 160 . . 3 (∀𝑥(𝜑𝑥 = 𝐴) → (∃𝑥𝜑𝐴 ∈ V))
10 eqeq2 2149 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
1110imbi2d 229 . . . . . . 7 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
1211albidv 1796 . . . . . 6 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
1312imbi1d 230 . . . . 5 (𝑦 = 𝐴 → ((∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)))
14 nfv 1508 . . . . . . 7 𝑦𝜑
1514mo2r 2051 . . . . . 6 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
161519.23bi 1571 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
1713, 16vtoclg 2746 . . . 4 (𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑))
1817com12 30 . . 3 (∀𝑥(𝜑𝑥 = 𝐴) → (𝐴 ∈ V → ∃*𝑥𝜑))
199, 18syld 45 . 2 (∀𝑥(𝜑𝑥 = 𝐴) → (∃𝑥𝜑 → ∃*𝑥𝜑))
20 moabs 2048 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
2119, 20sylibr 133 1 (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃*wmo 2000  Vcvv 2686 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688 This theorem is referenced by:  invdisj  3923
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