Theorem nsb 39150

Description: Generalization rule for negated wff. (Contributed by Steven Nguyen, 3-May-2023.)

Hypothesis

Ref	Expression
nsb.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
nsb	⊢ ¬ [𝑥 / 𝑦]𝜑

Proof of Theorem nsb

Step	Hyp	Ref	Expression
1		nsb.1	. . 3 ⊢ ¬ 𝜑
2	1	nex 1801	. 2 ⊢ ¬ ∃𝑦𝜑
3		spsbe 2088	. 2 ⊢ ([𝑥 / 𝑦]𝜑 → ∃𝑦𝜑)
4	2, 3	mto 199	1 ⊢ ¬ [𝑥 / 𝑦]𝜑

Syntax hints:  ¬ wn 3  ∃wex 1780  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970

This theorem depends on definitions:  df-bi 209  df-ex 1781  df-sb 2070

This theorem is referenced by:  sbn1  39151