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Theorem sbn1ALT 34812
Description: Alternate proof of sbn1 2111, 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
StepHypRef Expression
1 nsb 2110 . . . 4 (∀𝑥 ¬ (𝜑 ∧ ¬ 𝜑) → ¬ [𝑡 / 𝑥](𝜑 ∧ ¬ 𝜑))
2 pm3.24 406 . . . 4 ¬ (𝜑 ∧ ¬ 𝜑)
31, 2mpg 1805 . . 3 ¬ [𝑡 / 𝑥](𝜑 ∧ ¬ 𝜑)
4 sban 2088 . . 3 ([𝑡 / 𝑥](𝜑 ∧ ¬ 𝜑) ↔ ([𝑡 / 𝑥]𝜑 ∧ [𝑡 / 𝑥] ¬ 𝜑))
53, 4mtbi 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)))
75, 6mtoi 202 1 ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  [wsb 2072
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 2073
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator