Theorem sbn1 2144

Description: One direction of sbn 2315, using fewer axioms. Compare 19.2 2009. (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 2143	. . 3 ⊢ (∀𝑥 ¬ ⊥ → ¬ [𝑡 / 𝑥]⊥)
2		fal 1584	. . 3 ⊢ ¬ ⊥
3	1, 2	mpg 1830	. 2 ⊢ ¬ [𝑡 / 𝑥]⊥
4		pm2.21 124	. . 3 ⊢ (¬ 𝜑 → (𝜑 → ⊥))
5	4	sb2imi 2112	. 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 1582  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100

This theorem is used by:  bj-ab0  37638