Theorem aevlem0 2055

Description: Lemma for aevlem 2056. Instance of aev 2058. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-12 2178. (Revised by Wolf Lammen, 14-Mar-2021.) Extract from proof of a former lemma for axc11n 2431 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 2053	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2	1	alrimiv 1927	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
3		cbvaev 2054	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦)
4		equeuclr 2023	. . 3 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑦 → 𝑧 = 𝑥))
5	4	al2imi 1815	. 2 ⊢ (∀𝑧 𝑥 = 𝑦 → (∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑧 = 𝑥))
6	2, 3, 5	sylc 65	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)

Syntax hints:   → wi 4  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by:  aevlem  2056