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Theorem mo2icl 3659
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 variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq2 2748 . . . . . 6 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
21imbi2d 340 . . . . 5 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
32albidv 1922 . . . 4 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
43imbi1d 341 . . 3 (𝑦 = 𝐴 → ((∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)))
5 equequ2 2028 . . . . . . 7 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
65imbi2d 340 . . . . . 6 (𝑦 = 𝑧 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝑧)))
76albidv 1922 . . . . 5 (𝑦 = 𝑧 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝑧)))
8719.8aw 2052 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
9 df-mo 2538 . . . 4 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
108, 9sylibr 233 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
114, 10vtoclg 3514 . 2 (𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑))
12 eqvisset 3458 . . . . . 6 (𝑥 = 𝐴𝐴 ∈ V)
1312imim2i 16 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜑𝐴 ∈ V))
1413con3rr3 155 . . . 4 𝐴 ∈ V → ((𝜑𝑥 = 𝐴) → ¬ 𝜑))
1514alimdv 1918 . . 3 𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∀𝑥 ¬ 𝜑))
16 alnex 1782 . . . 4 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
17 nexmo 2539 . . . 4 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
1816, 17sylbi 216 . . 3 (∀𝑥 ¬ 𝜑 → ∃*𝑥𝜑)
1915, 18syl6 35 . 2 𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑))
2011, 19pm2.61i 182 1 (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1538   = wceq 1540  wex 1780  wcel 2105  ∃*wmo 2536  Vcvv 3441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443
This theorem is referenced by:  invdisj  5073  reusv1  5337  reusv2lem1  5338  opabiotafun  6899  fseqenlem2  9874  dfac2b  9979  imasaddfnlem  17328  imasvscafn  17337  bnj149  33095
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