Theorem aevlem0 2088

Description: Lemma for aevlem 2089. Instance of aev 2091. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-12 2212. (Revised by Wolf Lammen, 14-Mar-2021.) Extract from proof of a former lemma for axc11n 2455 and add DV condition to reduce axiom usage. (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

Step	Hyp	Ref	Expression
1		spaev 2086	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2	1	alrimiv 1959	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
3		cbvaev 2087	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦)
4		equeuclr 2055	. . 3 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑦 → 𝑧 = 𝑥))
5	4	al2imi 1847	. 2 ⊢ (∀𝑧 𝑥 = 𝑦 → (∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑧 = 𝑥))
6	2, 3, 5	sylc 66	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by:  aevlem  2089