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Theorem eliincex 40103
 Description: Counterexample to show that the additional conditions in eliin 4747 and eliin2 40109 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 5025 . . 3 ¬ V ∈ 𝑥𝐵 𝐶
31, 2eqneltri 40058 . 2 ¬ 𝐴 𝑥𝐵 𝐶
4 ral0 4300 . . 3 𝑥 ∈ ∅ 𝐴𝐶
5 eliinct.2 . . . 4 𝐵 = ∅
65raleqi 3354 . . 3 (∀𝑥𝐵 𝐴𝐶 ↔ ∀𝑥 ∈ ∅ 𝐴𝐶)
74, 6mpbir 223 . 2 𝑥𝐵 𝐴𝐶
8 pm3.22 453 . . . 4 ((¬ 𝐴 𝑥𝐵 𝐶 ∧ ∀𝑥𝐵 𝐴𝐶) → (∀𝑥𝐵 𝐴𝐶 ∧ ¬ 𝐴 𝑥𝐵 𝐶))
98olcd 905 . . 3 ((¬ 𝐴 𝑥𝐵 𝐶 ∧ ∀𝑥𝐵 𝐴𝐶) → ((𝐴 𝑥𝐵 𝐶 ∧ ¬ ∀𝑥𝐵 𝐴𝐶) ∨ (∀𝑥𝐵 𝐴𝐶 ∧ ¬ 𝐴 𝑥𝐵 𝐶)))
10 xor 1043 . . 3 (¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶) ↔ ((𝐴 𝑥𝐵 𝐶 ∧ ¬ ∀𝑥𝐵 𝐴𝐶) ∨ (∀𝑥𝐵 𝐴𝐶 ∧ ¬ 𝐴 𝑥𝐵 𝐶)))
119, 10sylibr 226 . 2 ((¬ 𝐴 𝑥𝐵 𝐶 ∧ ∀𝑥𝐵 𝐴𝐶) → ¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
123, 7, 11mp2an 683 1 ¬ (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 198   ∧ wa 386   ∨ wo 878   = wceq 1656   ∈ wcel 2164  ∀wral 3117  Vcvv 3414  ∅c0 4146  ∩ ciin 4743 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-v 3416  df-dif 3801  df-nul 4147 This theorem is referenced by: (None)
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