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Theorem dveeq1-o 34742
Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2445 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 1988 . 2 (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧)
2 ax-5 1988 . 2 (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧)
3 equequ1 2107 . 2 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
41, 2, 3dvelimf-o 34736 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-c5 34690  ax-c4 34691  ax-c7 34692  ax-c10 34693  ax-c11 34694  ax-c9 34697
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859
This theorem is referenced by:  ax12inda2ALT  34753
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