Theorem nsb 2142

Description: Any substitution in an always false formula is false. (Contributed by Steven Nguyen, 3-May-2023.)

Assertion

Ref	Expression
nsb	⊢ (∀𝑥 ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)

Proof of Theorem nsb

Step	Hyp	Ref	Expression
1		alnex 1813	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2	1	biimpi 219	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ¬ ∃𝑥𝜑)
3		spsbe 2118	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
4	2, 3	nsyl 141	1 ⊢ (∀𝑥 ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567  ∃wex 1811  [wsb 2098

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099

This theorem is used by:  sbn1  2143  noelOLD  4283  sbn1ALT  37866