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Theorem aev2 2054
Description: A version of aev 2053 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 2052, aev 2053, aev2 2054).

Using aev 2053 and alrimiv 1919, 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 28166. This list cannot be extended to ¬ or since the scheme 𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1787, ax-1 6-- ax-13 2381 (as the one-element universe shows).

(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 2053 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑤 𝑤 = 𝑠)
2 aev 2053 . . 3 (∀𝑤 𝑤 = 𝑠 → ∀𝑡 𝑢 = 𝑣)
32alrimiv 1919 . 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 1526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772
This theorem is referenced by:  hbaev  2055
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