Theorem nfsb4

Description: A variable not free in a proposition remains so after substitution in that proposition with a distinct variable (inference associated with nfsb4t ). Theorem nfsb replaces the distinctor antecedent with a disjoint variable condition. See nfsbv for a weaker version of nfsb not requiring ax-13 . (Contributed by NM, 14-May-1993) (Revised by Mario Carneiro, 4-Oct-2016) Usage of this theorem is discouraged because it depends on ax-13 . Use nfsbv instead. (New usage is discouraged.)

	Ref	Expression
Hypothesis	nfsb4.1	⊢ Ⅎ 𝑧 𝜑
Assertion	nfsb4	⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		nfsb4.1	⊢ Ⅎ 𝑧 𝜑
2		nfsb4t	⊢ ( ∀ 𝑥 Ⅎ 𝑧 𝜑 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ) )
3	2 1	mpg	⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 )