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Theorem dveeq2-o 36499
 Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2386 using ax-c15 36455. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dveeq2-o (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq2-o
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-5 1912 . 2 (𝑧 = 𝑤 → ∀𝑥 𝑧 = 𝑤)
2 ax-5 1912 . 2 (𝑧 = 𝑦 → ∀𝑤 𝑧 = 𝑦)
3 equequ2 2034 . 2 (𝑤 = 𝑦 → (𝑧 = 𝑤𝑧 = 𝑦))
41, 2, 3dvelimf-o 36495 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2380  ax-c5 36449  ax-c4 36450  ax-c7 36451  ax-c10 36452  ax-c11 36453  ax-c9 36456 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787 This theorem is referenced by:  ax12eq  36507  ax12el  36508  ax12inda  36514  ax12v2-o  36515
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