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Theorem mo2icl 2742
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 1450 . . . . 5 𝑥𝑥(𝜑𝑥 = 𝐴)
2 vex 2577 . . . . . . . 8 𝑥 ∈ V
3 eleq1 2116 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
42, 3mpbii 140 . . . . . . 7 (𝑥 = 𝐴𝐴 ∈ V)
54imim2i 12 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝜑𝐴 ∈ V))
65sps 1446 . . . . 5 (∀𝑥(𝜑𝑥 = 𝐴) → (𝜑𝐴 ∈ V))
71, 6eximd 1519 . . . 4 (∀𝑥(𝜑𝑥 = 𝐴) → (∃𝑥𝜑 → ∃𝑥 𝐴 ∈ V))
8 19.9v 1767 . . . 4 (∃𝑥 𝐴 ∈ V ↔ 𝐴 ∈ V)
97, 8syl6ib 154 . . 3 (∀𝑥(𝜑𝑥 = 𝐴) → (∃𝑥𝜑𝐴 ∈ V))
10 eqeq2 2065 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
1110imbi2d 223 . . . . . . 7 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
1211albidv 1721 . . . . . 6 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
1312imbi1d 224 . . . . 5 (𝑦 = 𝐴 → ((∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)))
14 nfv 1437 . . . . . . 7 𝑦𝜑
1514mo2r 1968 . . . . . 6 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
161519.23bi 1499 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
1713, 16vtoclg 2630 . . . 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 1965 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
2119, 20sylibr 141 1 (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257   = wceq 1259  wex 1397  wcel 1409  ∃*wmo 1917  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  invdisj  3786
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