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Theorem dveeq2-o 33019
Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2285 using ax-c15 32975. (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 1826 . 2 (𝑧 = 𝑤 → ∀𝑥 𝑧 = 𝑤)
2 ax-5 1826 . 2 (𝑧 = 𝑦 → ∀𝑤 𝑧 = 𝑦)
3 equequ2 1939 . 2 (𝑤 = 𝑦 → (𝑧 = 𝑤𝑧 = 𝑦))
41, 2, 3dvelimf-o 33015 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-c5 32969  ax-c4 32970  ax-c7 32971  ax-c10 32972  ax-c11 32973  ax-c9 32976
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700
This theorem is referenced by:  ax12eq  33027  ax12el  33028  ax12inda  33034  ax12v2-o  33035
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