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Theorem notzfaus 5354
Description: In the Separation Scheme zfauscl 5294, 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 5300 . . . . . . 7 ∅ ∈ V
32snnz 4775 . . . . . 6 {∅} ≠ ∅
41, 3eqnetri 3005 . . . . 5 𝐴 ≠ ∅
5 n0 4341 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
64, 5mpbi 229 . . . 4 𝑥 𝑥𝐴
7 pm5.19 386 . . . . 5 ¬ (𝑥𝑦 ↔ ¬ 𝑥𝑦)
8 ibar 528 . . . . . . 7 (𝑥𝐴 → (𝜑 ↔ (𝑥𝐴𝜑)))
9 notzfaus.2 . . . . . . 7 (𝜑 ↔ ¬ 𝑥𝑦)
108, 9bitr3di 286 . . . . . 6 (𝑥𝐴 → ((𝑥𝐴𝜑) ↔ ¬ 𝑥𝑦))
1110bibi2d 342 . . . . 5 (𝑥𝐴 → ((𝑥𝑦 ↔ (𝑥𝐴𝜑)) ↔ (𝑥𝑦 ↔ ¬ 𝑥𝑦)))
127, 11mtbiri 327 . . . 4 (𝑥𝐴 → ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)))
136, 12eximii 1831 . . 3 𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑))
14 exnal 1821 . . 3 (∃𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)) ↔ ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
1513, 14mpbi 229 . 2 ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
1615nex 1794 1 ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 395  wal 1531   = wceq 1533  wex 1773  wcel 2098  wne 2934  c0 4317  {csn 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-v 3470  df-dif 3946  df-nul 4318  df-sn 4624
This theorem is referenced by: (None)
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