Theorem sbc6

Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993) (Proof shortened by Eric Schmidt, 17-Jan-2007)

	Ref	Expression
Hypothesis	sbc6.1	⊢ 𝐴 ∈ V
Assertion	sbc6	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sbc6.1	⊢ 𝐴 ∈ V
2		sbc6g	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )
3	1 2	ax-mp	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) )