Theorem cbvralv

Description: Change the bound variable of a restricted universal quantifier using implicit substitution. See cbvralvw based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvralvw when possible. (Contributed by NM, 28-Jan-1997) (New usage is discouraged.)

	Ref	Expression
Hypothesis	cbvralv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvralv	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvralv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		nfv	⊢ Ⅎ 𝑦 𝜑
3		nfv	⊢ Ⅎ 𝑥 𝜓
4	2 3 1	cbvral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 𝜓 )