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Theorem dveeq1-o16 34540
 Description: Version of dveeq1 2336 using ax-c16 34496 instead of ax-5 1879. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1879. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dveeq1-o16 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq1-o16
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax5eq 34536 . 2 (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧)
2 ax5eq 34536 . 2 (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧)
3 equequ1 1998 . 2 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
41, 2, 3dvelimh 2367 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 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-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-c9 34494  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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