Theorem sbcn1

Description: Move negation in and out of class substitution. One direction of sbcng that holds for proper classes. (Contributed by NM, 17-Aug-2018)

	Ref	Expression
Assertion	sbcn1	⊢ ( [ 𝐴 / 𝑥 ] ¬ 𝜑 → ¬ [ 𝐴 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbcex	⊢ ( [ 𝐴 / 𝑥 ] ¬ 𝜑 → 𝐴 ∈ V )
2		sbcng	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] ¬ 𝜑 ↔ ¬ [ 𝐴 / 𝑥 ] 𝜑 ) )
3	2	biimpd	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] ¬ 𝜑 → ¬ [ 𝐴 / 𝑥 ] 𝜑 ) )
4	1 3	mpcom	⊢ ( [ 𝐴 / 𝑥 ] ¬ 𝜑 → ¬ [ 𝐴 / 𝑥 ] 𝜑 )