Theorem aevlem 2056

Description: Lemma for aev 2058 and axc16g 2250. Change free and bound variables. Instance of aev 2058. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2370, 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.

Step	Hyp	Ref	Expression
1		cbvaev 2054	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑦)
2		aevlem0 2055	. 2 ⊢ (∀𝑢 𝑢 = 𝑦 → ∀𝑥 𝑥 = 𝑢)
3		cbvaev 2054	. 2 ⊢ (∀𝑥 𝑥 = 𝑢 → ∀𝑡 𝑡 = 𝑢)
4		aevlem0 2055	. 2 ⊢ (∀𝑡 𝑡 = 𝑢 → ∀𝑧 𝑧 = 𝑡)
5	1, 2, 3, 4	4syl 19	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡)

Syntax hints:   → wi 4  ∀wal 1537

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 1911  ax-6 1969  ax-7 2009

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1780

This theorem is referenced by:  aeveq  2057  aev  2058  axc16g  2250  bj-axc16g16  34911  bj-axc11nv  35034  bj-aecomsv  35035