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Theorem aevlem 1983
 Description: Lemma for aev 1985 and axc16g 2135. Change free and bound variables. Instance of aev 1985. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2250, 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 1981 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑦)
2 aevlem0 1982 . 2 (∀𝑢 𝑢 = 𝑦 → ∀𝑥 𝑥 = 𝑢)
3 cbvaev 1981 . 2 (∀𝑥 𝑥 = 𝑢 → ∀𝑡 𝑡 = 𝑢)
4 aevlem0 1982 . 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 1478 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702 This theorem is referenced by:  aeveq  1984  aev  1985  hbaevg  1986  axc16g  2135  axc11vOLD  2142  axc16gOLD  2163  aevOLD  2164  aevALTOLD  2325  bj-axc16g16  32308  bj-axc11nv  32380
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