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Theorem notzfaus 5254
Description: In the Separation Scheme zfauscl 5197, 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 if that requirement is not met. (Contributed by NM, 8-Feb-2006.) (Proof shortened by BJ, 18-Nov-2023.)
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 5203 . . . . . . 7 ∅ ∈ V
32snnz 4705 . . . . . 6 {∅} ≠ ∅
41, 3eqnetri 3086 . . . . 5 𝐴 ≠ ∅
5 n0 4309 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
64, 5mpbi 231 . . . 4 𝑥 𝑥𝐴
7 pm5.19 388 . . . . 5 ¬ (𝑥𝑦 ↔ ¬ 𝑥𝑦)
8 notzfaus.2 . . . . . . 7 (𝜑 ↔ ¬ 𝑥𝑦)
9 ibar 529 . . . . . . 7 (𝑥𝐴 → (𝜑 ↔ (𝑥𝐴𝜑)))
108, 9syl5rbbr 287 . . . . . 6 (𝑥𝐴 → ((𝑥𝐴𝜑) ↔ ¬ 𝑥𝑦))
1110bibi2d 344 . . . . 5 (𝑥𝐴 → ((𝑥𝑦 ↔ (𝑥𝐴𝜑)) ↔ (𝑥𝑦 ↔ ¬ 𝑥𝑦)))
127, 11mtbiri 328 . . . 4 (𝑥𝐴 → ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)))
136, 12eximii 1828 . . 3 𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑))
14 exnal 1818 . . 3 (∃𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)) ↔ ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
1513, 14mpbi 231 . 2 ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
1615nex 1792 1 ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396  wal 1526   = wceq 1528  wex 1771  wcel 2105  wne 3016  c0 4290  {csn 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-nul 5202
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3497  df-dif 3938  df-nul 4291  df-sn 4560
This theorem is referenced by: (None)
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