Metamath Proof Explorer


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 ( ( 𝜑𝑥 = 𝑦 ) → ( 𝜓𝜒 ) )
Assertion cbvraldva ( 𝜑 → ( ∀ 𝑥𝐴 𝜓 ↔ ∀ 𝑦𝐴 𝜒 ) )

Proof

Step Hyp Ref Expression
1 cbvraldva.1 ( ( 𝜑𝑥 = 𝑦 ) → ( 𝜓𝜒 ) )
2 eqidd ( ( 𝜑𝑥 = 𝑦 ) → 𝐴 = 𝐴 )
3 1 2 cbvraldva2 ( 𝜑 → ( ∀ 𝑥𝐴 𝜓 ↔ ∀ 𝑦𝐴 𝜒 ) )