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Theorem dveel2ALT 34543
 Description: Alternate proof of dveel2 2399 using ax-c16 34496 instead of ax-5 1879. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dveel2ALT (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveel2ALT
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax5el 34541 . 2 (𝑧𝑤 → ∀𝑥 𝑧𝑤)
2 ax5el 34541 . 2 (𝑧𝑦 → ∀𝑤 𝑧𝑦)
3 elequ2 2044 . 2 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
41, 2, 3dvelimh 2367 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-c14 34495  ax-c16 34496 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750 This theorem is referenced by: (None)
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