Description: A version of aev 2060
with two universal quantifiers in the consequent.
One can prove similar statements with arbitrary numbers of universal
quantifiers in the consequent (the series begins with aeveq 2059, aev 2060,
aev2 2061).
Using aev 2060 and alrimiv 1930, one can actually prove (with no more axioms)
any scheme of the form (∀𝑥𝑥 = 𝑦 → PHI) , DV (𝑥, 𝑦) where
PHI involves only setvar variables and the connectors →, ↔,
∧, ∨, ⊤, =, ∀, ∃, ∃*, ∃!,
Ⅎ. An example is given by aevdemo 28824. This list cannot be
extended to ¬ or ⊥ since the scheme ∀𝑥𝑥 = 𝑦 is
consistent with ax-mp 5, ax-gen 1798, ax-1 6--
ax-13 2372 (as the
one-element universe shows), so for instance (∀𝑥𝑥 = 𝑦 → ⊥),
DV (𝑥, 𝑦) is not provable from these axioms
alone (indeed, dtru 5359
uses non-logical axioms as well). (Contributed by BJ,
23-Mar-2021.) |