Theorem hbaevg 2101

Description: Generalization of hbaev 2102, proved at no extra cost. Instance of aev2 2105. (Contributed by Wolf Lammen, 22-Mar-2021.) (Revised by BJ, 29-Mar-2021.)

Assertion

Ref	Expression
hbaevg	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑡 𝑡 = 𝑢)

Distinct variable groups:   𝑥,𝑦   𝑢,𝑡

Proof of Theorem hbaevg

Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aevlem 2098	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑣 𝑣 = 𝑤)
2		aevlem 2098	. . 3 ⊢ (∀𝑣 𝑣 = 𝑤 → ∀𝑡 𝑡 = 𝑢)
3	2	alrimiv 1970	. 2 ⊢ (∀𝑣 𝑣 = 𝑤 → ∀𝑧∀𝑡 𝑡 = 𝑢)
4	1, 3	syl 17	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑡 𝑡 = 𝑢)

Syntax hints:   → wi 4  ∀wal 1599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824

This theorem is referenced by:  hbaev  2102  aev2  2105