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Theorem notzfaus 4800
Description: In the Separation Scheme zfauscl 4743, 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 4750 . . . . . . 7 ∅ ∈ V
32snnz 4279 . . . . . 6 {∅} ≠ ∅
41, 3eqnetri 2860 . . . . 5 𝐴 ≠ ∅
5 n0 3907 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
64, 5mpbi 220 . . . 4 𝑥 𝑥𝐴
7 biimt 350 . . . . . 6 (𝑥𝐴 → (𝑥𝑦 ↔ (𝑥𝐴𝑥𝑦)))
8 iman 440 . . . . . . 7 ((𝑥𝐴𝑥𝑦) ↔ ¬ (𝑥𝐴 ∧ ¬ 𝑥𝑦))
9 notzfaus.2 . . . . . . . 8 (𝜑 ↔ ¬ 𝑥𝑦)
109anbi2i 729 . . . . . . 7 ((𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝑦))
118, 10xchbinxr 325 . . . . . 6 ((𝑥𝐴𝑥𝑦) ↔ ¬ (𝑥𝐴𝜑))
127, 11syl6bb 276 . . . . 5 (𝑥𝐴 → (𝑥𝑦 ↔ ¬ (𝑥𝐴𝜑)))
13 xor3 372 . . . . 5 (¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)) ↔ (𝑥𝑦 ↔ ¬ (𝑥𝐴𝜑)))
1412, 13sylibr 224 . . . 4 (𝑥𝐴 → ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)))
156, 14eximii 1761 . . 3 𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑))
16 exnal 1751 . . 3 (∃𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)) ↔ ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
1715, 16mpbi 220 . 2 ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
1817nex 1728 1 ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  wne 2790  c0 3891  {csn 4148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3188  df-dif 3558  df-nul 3892  df-sn 4149
This theorem is referenced by: (None)
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