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Metamath Proof Explorer

Theorem nfsb2

Description: Bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Mario Carneiro, 4-Oct-2016) (New usage is discouraged.)

		Ref	Expression
	Assertion	nfsb2	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		nfna1	⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑦
2		hbsb2	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 ) )
3	1 2	nf5d	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )