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Theorem dveeq1-o 35951
Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2389 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 1902 . 2 (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧)
2 ax-5 1902 . 2 (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧)
3 equequ1 2023 . 2 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
41, 2, 3dvelimf-o 35945 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
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:  ax12inda2ALT  35962
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