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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbn1ALT | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
sbn1ALT | ⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsb 2110 | . . . 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 2088 | . . 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) |
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) |
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