Theorem sbn1 2141

Description: One direction of sbn 2314, using fewer axioms. Compare 19.2 2005. (Contributed by Steven Nguyen, 18-Aug-2023.)

Assertion

Ref	Expression
sbn1	⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)

Proof of Theorem sbn1

Step	Hyp	Ref	Expression
1		nsb 2140	. . 3 ⊢ (∀𝑥 ¬ ⊥ → ¬ [𝑡 / 𝑥]⊥)
2		fal 1583	. . 3 ⊢ ¬ ⊥
3	1, 2	mpg 1826	. 2 ⊢ ¬ [𝑡 / 𝑥]⊥
4		pm2.21 124	. . 3 ⊢ (¬ 𝜑 → (𝜑 → ⊥))
5	4	sb2imi 2108	. 2 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]⊥))
6	3, 5	mtoi 202	1 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ⊥wfal 1581  [wsb 2095

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096

This theorem is used by:  bj-ab0  37571