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Theorem aevlem0 2123
Description: Lemma for aevlem 2124. Instance of aev 2126. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-12 2188. (Revised by Wolf Lammen, 14-Mar-2021.) (Revised by BJ, 29-Mar-2021.) (Proof shortened by Wolf Lammen, 30-Mar-2021.)
Assertion
Ref Expression
aevlem0 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem aevlem0
StepHypRef Expression
1 spaev 2121 . . 3 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
21alrimiv 1996 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
3 cbvaev 2122 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦)
4 equeuclr 2097 . . 3 (𝑥 = 𝑦 → (𝑧 = 𝑦𝑧 = 𝑥))
54al2imi 1884 . 2 (∀𝑧 𝑥 = 𝑦 → (∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑧 = 𝑥))
62, 3, 5sylc 65 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1846
This theorem is referenced by:  aevlem  2124
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