Theorem sbn1ALT 34736

Description: Alternate proof of sbn1 2109, 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 2108	. . . 4 ⊢ (∀𝑥 ¬ (𝜑 ∧ ¬ 𝜑) → ¬ [𝑡 / 𝑥](𝜑 ∧ ¬ 𝜑))
2		pm3.24 406	. . . 4 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)
3	1, 2	mpg 1805	. . 3 ⊢ ¬ [𝑡 / 𝑥](𝜑 ∧ ¬ 𝜑)
4		sban 2086	. . 3 ⊢ ([𝑡 / 𝑥](𝜑 ∧ ¬ 𝜑) ↔ ([𝑡 / 𝑥]𝜑 ∧ [𝑡 / 𝑥] ¬ 𝜑))
5	3, 4	mtbi 325	. 2 ⊢ ¬ ([𝑡 / 𝑥]𝜑 ∧ [𝑡 / 𝑥] ¬ 𝜑)
6		pm3.21 475	. 2 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ([𝑡 / 𝑥]𝜑 → ([𝑡 / 𝑥]𝜑 ∧ [𝑡 / 𝑥] ¬ 𝜑)))
7	5, 6	mtoi 202	1 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  [wsb 2070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2071

This theorem is referenced by: (None)