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Theorem dveeq2-o 38405
Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2373 using ax-c15 38361. (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 1906 . 2 (𝑧 = 𝑤 → ∀𝑥 𝑧 = 𝑤)
2 ax-5 1906 . 2 (𝑧 = 𝑦 → ∀𝑤 𝑧 = 𝑦)
3 equequ2 2022 . 2 (𝑤 = 𝑦 → (𝑧 = 𝑤𝑧 = 𝑦))
41, 2, 3dvelimf-o 38401 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2367  ax-c5 38355  ax-c4 38356  ax-c7 38357  ax-c10 38358  ax-c11 38359  ax-c9 38362
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779
This theorem is referenced by:  ax12eq  38413  ax12el  38414  ax12inda  38420  ax12v2-o  38421
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