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Theorem aevlem 2051
Description: Lemma for aev 2053 and axc16g 2251. Change free and bound variables. Instance of aev 2053. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2381, 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 2049 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑦)
2 aevlem0 2050 . 2 (∀𝑢 𝑢 = 𝑦 → ∀𝑥 𝑥 = 𝑢)
3 cbvaev 2049 . 2 (∀𝑥 𝑥 = 𝑢 → ∀𝑡 𝑡 = 𝑢)
4 aevlem0 2050 . 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 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:  aeveq  2052  aev  2053  axc16g  2251  bj-axc16g16  33915  bj-axc11nv  34026  bj-aecomsv  34027
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