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Theorem dveeq2-o 37395
Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2376 using ax-c15 37351. (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 1913 . 2 (𝑧 = 𝑤 → ∀𝑥 𝑧 = 𝑤)
2 ax-5 1913 . 2 (𝑧 = 𝑦 → ∀𝑤 𝑧 = 𝑦)
3 equequ2 2029 . 2 (𝑤 = 𝑦 → (𝑧 = 𝑤𝑧 = 𝑦))
41, 2, 3dvelimf-o 37391 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1539
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 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2370  ax-c5 37345  ax-c4 37346  ax-c7 37347  ax-c10 37348  ax-c11 37349  ax-c9 37352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786
This theorem is referenced by:  ax12eq  37403  ax12el  37404  ax12inda  37410  ax12v2-o  37411
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