Theorem aevlem 2080

Description: Lemma for aev 2082 and axc16g 2298. Change free and bound variables. Instance of aev 2082. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2406, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Reduce axiom usage. (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.

Step	Hyp	Ref	Expression
1		cbvaev 2078	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑦)
2		aevlem0 2079	. 2 ⊢ (∀𝑢 𝑢 = 𝑦 → ∀𝑥 𝑥 = 𝑢)
3		cbvaev 2078	. 2 ⊢ (∀𝑥 𝑥 = 𝑢 → ∀𝑡 𝑡 = 𝑢)
4		aevlem0 2079	. 2 ⊢ (∀𝑡 𝑡 = 𝑢 → ∀𝑧 𝑧 = 𝑡)
5	1, 2, 3, 4	4syl 20	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡)

Syntax hints:   → wi 4  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by:  aeveq  2081  aev  2082  axc16g  2298  bj-axc16g16  37171  bj-axc11nv  37304  bj-aecomsv  37305