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Theorem notzfaus 4766
 Description: In the Separation Scheme zfauscl 4711, we require that 𝑦 not occur in 𝜑 (which can be generalized to "not be free in"). Here we show special cases of 𝐴 and 𝜑 that result in a contradiction by violating this requirement. (Contributed by NM, 8-Feb-2006.)
Hypotheses
Ref Expression
notzfaus.1 𝐴 = {∅}
notzfaus.2 (𝜑 ↔ ¬ 𝑥𝑦)
Assertion
Ref Expression
notzfaus ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)

Proof of Theorem notzfaus
StepHypRef Expression
1 notzfaus.1 . . . . . 6 𝐴 = {∅}
2 0ex 4718 . . . . . . 7 ∅ ∈ V
32snnz 4252 . . . . . 6 {∅} ≠ ∅
41, 3eqnetri 2852 . . . . 5 𝐴 ≠ ∅
5 n0 3890 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
64, 5mpbi 219 . . . 4 𝑥 𝑥𝐴
7 biimt 349 . . . . . 6 (𝑥𝐴 → (𝑥𝑦 ↔ (𝑥𝐴𝑥𝑦)))
8 iman 439 . . . . . . 7 ((𝑥𝐴𝑥𝑦) ↔ ¬ (𝑥𝐴 ∧ ¬ 𝑥𝑦))
9 notzfaus.2 . . . . . . . 8 (𝜑 ↔ ¬ 𝑥𝑦)
109anbi2i 726 . . . . . . 7 ((𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝑦))
118, 10xchbinxr 324 . . . . . 6 ((𝑥𝐴𝑥𝑦) ↔ ¬ (𝑥𝐴𝜑))
127, 11syl6bb 275 . . . . 5 (𝑥𝐴 → (𝑥𝑦 ↔ ¬ (𝑥𝐴𝜑)))
13 xor3 371 . . . . 5 (¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)) ↔ (𝑥𝑦 ↔ ¬ (𝑥𝐴𝜑)))
1412, 13sylibr 223 . . . 4 (𝑥𝐴 → ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)))
156, 14eximii 1754 . . 3 𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑))
16 exnal 1744 . . 3 (∃𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)) ↔ ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
1715, 16mpbi 219 . 2 ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
1817nex 1722 1 ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-nul 3875  df-sn 4126 This theorem is referenced by: (None)
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