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Theorem notzfaus 5362
Description: In the Separation Scheme zfauscl 5297, 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 5306 . . . . . . 7 ∅ ∈ V
32snnz 4775 . . . . . 6 {∅} ≠ ∅
41, 3eqnetri 3010 . . . . 5 𝐴 ≠ ∅
5 n0 4352 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
64, 5mpbi 230 . . . 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 1836 . . 3 𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑))
14 exnal 1826 . . 3 (∃𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)) ↔ ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
1513, 14mpbi 230 . 2 ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
1615nex 1799 1 ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wal 1537   = wceq 1539  wex 1778  wcel 2107  wne 2939  c0 4332  {csn 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-v 3481  df-dif 3953  df-nul 4333  df-sn 4626
This theorem is referenced by: (None)
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