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Theorem aev2 2053
Description: A version of aev 2052 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 2051, aev 2052, aev2 2053).

Using aev 2052 and alrimiv 1922, 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 , , , , , =, , , ∃*, ∃!, . An example is given by aevdemo 30182. This list cannot be extended to ¬ or since the scheme 𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1789, ax-1 6-- ax-13 2363 (as the one-element universe shows), so for instance (∀𝑥𝑥 = 𝑦 → ⊥), DV (𝑥, 𝑦) is not provable from these axioms alone (indeed, dtru 5426 uses non-logical axioms as well). (Contributed by BJ, 23-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 aev 2052 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑤 𝑤 = 𝑠)
2 aev 2052 . . 3 (∀𝑤 𝑤 = 𝑠 → ∀𝑡 𝑢 = 𝑣)
32alrimiv 1922 . 2 (∀𝑤 𝑤 = 𝑠 → ∀𝑧𝑡 𝑢 = 𝑣)
41, 3syl 17 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑡 𝑢 = 𝑣)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774
This theorem is referenced by:  hbaev  2054
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