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Theorem aevlem 2023
Description: Lemma for aev 2025 and axc16g 2172. Change free and bound variables. Instance of aev 2025. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2282, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 29-Mar-2021.)
Assertion
Ref Expression
aevlem (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡)
Distinct variable groups:   𝑥,𝑦   𝑧,𝑡

Proof of Theorem aevlem
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 cbvaev 2021 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑦)
2 aevlem0 2022 . 2 (∀𝑢 𝑢 = 𝑦 → ∀𝑥 𝑥 = 𝑢)
3 cbvaev 2021 . 2 (∀𝑥 𝑥 = 𝑢 → ∀𝑡 𝑡 = 𝑢)
4 aevlem0 2022 . 2 (∀𝑡 𝑡 = 𝑢 → ∀𝑧 𝑧 = 𝑡)
51, 2, 3, 44syl 19 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  aeveq  2024  aev  2025  hbaevg  2026  axc16g  2172  axc11vOLD  2179  axc16gOLD  2198  bj-axc16g16  32799  bj-axc11nv  32870
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