Theorem cbvraldva

Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017)

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

Proof

Step	Hyp	Ref	Expression
1		cbvraldva.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
2		eqidd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐴 )
3	1 2	cbvraldva2	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑦 ∈ 𝐴 𝜒 ) )