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Theorem dveeq2-o 35949
Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2387 using ax-c15 35905. (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 1902 . 2 (𝑧 = 𝑤 → ∀𝑥 𝑧 = 𝑤)
2 ax-5 1902 . 2 (𝑧 = 𝑦 → ∀𝑤 𝑧 = 𝑦)
3 equequ2 2024 . 2 (𝑤 = 𝑦 → (𝑧 = 𝑤𝑧 = 𝑦))
41, 2, 3dvelimf-o 35945 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-c5 35899  ax-c4 35900  ax-c7 35901  ax-c10 35902  ax-c11 35903  ax-c9 35906
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776
This theorem is referenced by:  ax12eq  35957  ax12el  35958  ax12inda  35964  ax12v2-o  35965
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