Theorem cbvraldvaOLD

Description: Obsolete version of cbvraldva as of 9-Mar-2025. (Contributed by David Moews, 1-May-2017) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

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