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Theorem dveeq1-o 33238
Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2288 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 1827 . 2 (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧)
2 ax-5 1827 . 2 (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧)
3 equequ1 1939 . 2 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
41, 2, 3dvelimf-o 33232 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-c5 33186  ax-c4 33187  ax-c7 33188  ax-c10 33189  ax-c11 33190  ax-c9 33193
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701
This theorem is referenced by:  ax12inda2ALT  33249
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