Theorem sbn1ALT 37281

Description: Alternate proof of sbn1 2131, not using the false constant. (Contributed by BJ, 18-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sbn1ALT	⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)

Proof of Theorem sbn1ALT

Step	Hyp	Ref	Expression
1		nsb 2130	. . . 4 ⊢ (∀𝑥 ¬ (𝜑 ∧ ¬ 𝜑) → ¬ [𝑡 / 𝑥](𝜑 ∧ ¬ 𝜑))
2		pm3.24 405	. . . 4 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)
3	1, 2	mpg 1807	. . 3 ⊢ ¬ [𝑡 / 𝑥](𝜑 ∧ ¬ 𝜑)
4		sban 2103	. . 3 ⊢ ([𝑡 / 𝑥](𝜑 ∧ ¬ 𝜑) ↔ ([𝑡 / 𝑥]𝜑 ∧ [𝑡 / 𝑥] ¬ 𝜑))
5	3, 4	mtbi 324	. 2 ⊢ ¬ ([𝑡 / 𝑥]𝜑 ∧ [𝑡 / 𝑥] ¬ 𝜑)
6		pm3.21 474	. 2 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ([𝑡 / 𝑥]𝜑 → ([𝑡 / 𝑥]𝜑 ∧ [𝑡 / 𝑥] ¬ 𝜑)))
7	5, 6	mtoi 201	1 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398  [wsb 2080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1790  df-sb 2081

This theorem is referenced by: (None)