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Theorem mo2icl 2931
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 1552 . . . . 5 𝑥𝑥(𝜑𝑥 = 𝐴)
2 vex 2755 . . . . . . . 8 𝑥 ∈ V
3 eleq1 2252 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
42, 3mpbii 148 . . . . . . 7 (𝑥 = 𝐴𝐴 ∈ V)
54imim2i 12 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝜑𝐴 ∈ V))
65sps 1548 . . . . 5 (∀𝑥(𝜑𝑥 = 𝐴) → (𝜑𝐴 ∈ V))
71, 6eximd 1623 . . . 4 (∀𝑥(𝜑𝑥 = 𝐴) → (∃𝑥𝜑 → ∃𝑥 𝐴 ∈ V))
8 19.9v 1882 . . . 4 (∃𝑥 𝐴 ∈ V ↔ 𝐴 ∈ V)
97, 8imbitrdi 161 . . 3 (∀𝑥(𝜑𝑥 = 𝐴) → (∃𝑥𝜑𝐴 ∈ V))
10 eqeq2 2199 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
1110imbi2d 230 . . . . . . 7 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
1211albidv 1835 . . . . . 6 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
1312imbi1d 231 . . . . 5 (𝑦 = 𝐴 → ((∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)))
14 nfv 1539 . . . . . . 7 𝑦𝜑
1514mo2r 2090 . . . . . 6 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
161519.23bi 1603 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
1713, 16vtoclg 2812 . . . 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 2087 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
2119, 20sylibr 134 1 (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362   = wceq 1364  wex 1503  ∃*wmo 2039  wcel 2160  Vcvv 2752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754
This theorem is referenced by:  invdisj  4012  imasaddfnlemg  12788
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