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Theorem nfeqf2 2387
Description: An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2382. (Contributed by Wolf Lammen, 9-Jun-2019.) Remove dependency on ax-12 2176. (Revised by Wolf Lammen, 16-Dec-2022.) (New usage is discouraged.)
Assertion
Ref Expression
nfeqf2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
Distinct variable group:   𝑥,𝑧

Proof of Theorem nfeqf2
StepHypRef Expression
1 exnal 1828 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2 hbe1 2145 . . . . 5 (∃𝑥 𝑧 = 𝑦 → ∀𝑥𝑥 𝑧 = 𝑦)
3 ax13lem2 2386 . . . . . 6 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))
4 ax13lem1 2384 . . . . . 6 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
53, 4syldc 48 . . . . 5 (∃𝑥 𝑧 = 𝑦 → (¬ 𝑥 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
62, 5eximdh 1865 . . . 4 (∃𝑥 𝑧 = 𝑦 → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥𝑥 𝑧 = 𝑦))
7 hbe1a 2146 . . . 4 (∃𝑥𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)
86, 7syl6com 37 . . 3 (∃𝑥 ¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
98nfd 1792 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
101, 9sylbir 238 1 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1536  wex 1781  wnf 1785
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-10 2143  ax-13 2382
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786
This theorem is referenced by:  dveeq2  2388  nfeqf1  2389  sb4b  2491  sb4bOLD  2492  sbal1  2551  copsexg  5350  axrepndlem1  10007  axpowndlem2  10013  axpowndlem3  10014  bj-dvelimdv  34291  bj-dvelimdv1  34292  wl-equsb3  34956  wl-sbcom2d-lem1  34959  wl-mo2df  34970  wl-eudf  34972  wl-euequf  34974
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