Theorem ceqsal

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	ceqsal.1	⊢ Ⅎ 𝑥 𝜓
	ceqsal.2	⊢ 𝐴 ∈ V
	ceqsal.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ceqsal	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ceqsal.1	⊢ Ⅎ 𝑥 𝜓
2		ceqsal.2	⊢ 𝐴 ∈ V
3		ceqsal.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
4	1 3	ceqsalg	⊢ ( 𝐴 ∈ V → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )
5	2 4	ax-mp	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 )