Theorem ceqsalv

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993)

	Ref	Expression
Hypotheses	ceqsalv.1	⊢ 𝐴 ∈ V
	ceqsalv.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ceqsalv	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ceqsalv.1	⊢ 𝐴 ∈ V
2		ceqsalv.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		nfv	⊢ Ⅎ 𝑥 𝜓
4	3 1 2	ceqsal	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 )