Theorem nsb 2109

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 1782	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2	1	biimpi 216	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ¬ ∃𝑥𝜑)
3		spsbe 2085	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
4	2, 3	nsyl 140	1 ⊢ (∀𝑥 ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539  ∃wex 1780  [wsb 2067

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 1968

This theorem depends on definitions:  df-bi 207  df-ex 1781  df-sb 2068

This theorem is referenced by:  sbn1  2110  noel  4285  sbn1ALT  36900