Theorem cbvaev 2085

Description: Change bound variable in an equality with a disjoint variable condition. Instance of aev 2089. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.)

Assertion

Ref	Expression
cbvaev	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦)

Distinct variable groups:   𝑥,𝑦   𝑦,𝑧

Proof of Theorem cbvaev

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax7 2046	. . 3 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑦 → 𝑡 = 𝑦))
2	1	cbvalivw 2037	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑡 𝑡 = 𝑦)
3		ax7 2046	. . 3 ⊢ (𝑡 = 𝑧 → (𝑡 = 𝑦 → 𝑧 = 𝑦))
4	3	cbvalivw 2037	. 2 ⊢ (∀𝑡 𝑡 = 𝑦 → ∀𝑧 𝑧 = 𝑦)
5	2, 4	syl 18	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦)

Syntax hints:   → wi 4  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is referenced by:  aevlem0  2086  aevlem  2087