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Theorem eliincex 38777
 Description: Counterexample to show that the additional conditions in eliin 4491 and eliin2 38783 are actually needed. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
eliinct.1 𝐴 = V
eliinct.2 𝐵 = ∅
Assertion
Ref Expression
eliincex ¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem eliincex
StepHypRef Expression
1 eliinct.1 . . 3 𝐴 = V
2 nvel 4757 . . 3 ¬ V ∈ 𝑥𝐵 𝐶
31, 2eqneltri 38728 . 2 ¬ 𝐴 𝑥𝐵 𝐶
4 ral0 4048 . . 3 𝑥 ∈ ∅ 𝐴𝐶
5 eliinct.2 . . . 4 𝐵 = ∅
65raleqi 3131 . . 3 (∀𝑥𝐵 𝐴𝐶 ↔ ∀𝑥 ∈ ∅ 𝐴𝐶)
74, 6mpbir 221 . 2 𝑥𝐵 𝐴𝐶
8 pm3.22 465 . . . 4 ((¬ 𝐴 𝑥𝐵 𝐶 ∧ ∀𝑥𝐵 𝐴𝐶) → (∀𝑥𝐵 𝐴𝐶 ∧ ¬ 𝐴 𝑥𝐵 𝐶))
98olcd 408 . . 3 ((¬ 𝐴 𝑥𝐵 𝐶 ∧ ∀𝑥𝐵 𝐴𝐶) → ((𝐴 𝑥𝐵 𝐶 ∧ ¬ ∀𝑥𝐵 𝐴𝐶) ∨ (∀𝑥𝐵 𝐴𝐶 ∧ ¬ 𝐴 𝑥𝐵 𝐶)))
10 xor 934 . . 3 (¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶) ↔ ((𝐴 𝑥𝐵 𝐶 ∧ ¬ ∀𝑥𝐵 𝐴𝐶) ∨ (∀𝑥𝐵 𝐴𝐶 ∧ ¬ 𝐴 𝑥𝐵 𝐶)))
119, 10sylibr 224 . 2 ((¬ 𝐴 𝑥𝐵 𝐶 ∧ ∀𝑥𝐵 𝐴𝐶) → ¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
123, 7, 11mp2an 707 1 ¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  Vcvv 3186  ∅c0 3891  ∩ ciin 4486 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3188  df-dif 3558  df-nul 3892 This theorem is referenced by: (None)
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