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Theorem notzfaus 5230
 Description: In the Separation Scheme zfauscl 5172, 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 5178 . . . . . . 7 ∅ ∈ V
32snnz 4675 . . . . . 6 {∅} ≠ ∅
41, 3eqnetri 3060 . . . . 5 𝐴 ≠ ∅
5 n0 4263 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
64, 5mpbi 233 . . . 4 𝑥 𝑥𝐴
7 pm5.19 391 . . . . 5 ¬ (𝑥𝑦 ↔ ¬ 𝑥𝑦)
8 ibar 532 . . . . . . 7 (𝑥𝐴 → (𝜑 ↔ (𝑥𝐴𝜑)))
9 notzfaus.2 . . . . . . 7 (𝜑 ↔ ¬ 𝑥𝑦)
108, 9bitr3di 289 . . . . . 6 (𝑥𝐴 → ((𝑥𝐴𝜑) ↔ ¬ 𝑥𝑦))
1110bibi2d 346 . . . . 5 (𝑥𝐴 → ((𝑥𝑦 ↔ (𝑥𝐴𝜑)) ↔ (𝑥𝑦 ↔ ¬ 𝑥𝑦)))
127, 11mtbiri 330 . . . 4 (𝑥𝐴 → ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)))
136, 12eximii 1838 . . 3 𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑))
14 exnal 1828 . . 3 (∃𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)) ↔ ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
1513, 14mpbi 233 . 2 ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
1615nex 1802 1 ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2112   ≠ wne 2990  ∅c0 4246  {csn 4528 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773  ax-nul 5177 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ne 2991  df-v 3446  df-dif 3887  df-nul 4247  df-sn 4529 This theorem is referenced by: (None)
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