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Theorem aev2 1973
 Description: A version of aev 1970 with two universal quantifiers in the consequent, and a generalization of hbaevg 1971. One can prove similar statements with arbitrary numbers of universal quantifiers in the consequent (the series begins with aeveq 1969, aev 1970, aev2 1973). Using aev 1970 and alrimiv 1842 (as in aev2ALT 1974), 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 26709. This list cannot be extended to ¬ or ⊥ since the scheme ∀𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1713, ax-1 6-- ax-13 2234 (as the one-element universe shows). (Contributed by BJ, 29-Mar-2021.)
Assertion
Ref Expression
aev2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑡 𝑢 = 𝑣)
Distinct variable group:   𝑥,𝑦

Proof of Theorem aev2
Dummy variables 𝑤 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbaevg 1971 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑤 𝑤 = 𝑠)
2 aev 1970 . 2 (∀𝑤 𝑤 = 𝑠 → ∀𝑡 𝑢 = 𝑣)
31, 2sylg 1740 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑡 𝑢 = 𝑣)
 Colors of variables: wff setvar class Syntax hints:   → 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 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by: (None)
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