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Theorem dveeq1-o 36511
 Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2387 using ax-c11 . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dveeq1-o (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq1-o
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-5 1911 . 2 (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧)
2 ax-5 1911 . 2 (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧)
3 equequ1 2032 . 2 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
41, 2, 3dvelimf-o 36505 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536 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 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-c5 36459  ax-c4 36460  ax-c7 36461  ax-c10 36462  ax-c11 36463  ax-c9 36466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786 This theorem is referenced by:  ax12inda2ALT  36522
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