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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbn1 | Structured version Visualization version GIF version |
Description: One direction of sbn 2283, using fewer axioms. Compare 19.2 1981. (Contributed by Steven Nguyen, 18-Aug-2023.) |
Ref | Expression |
---|---|
sbn1 | ⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fal 1552 | . . 3 ⊢ ¬ ⊥ | |
2 | 1 | nsb 39392 | . 2 ⊢ ¬ [𝑡 / 𝑥]⊥ |
3 | pm2.21 123 | . . 3 ⊢ (¬ 𝜑 → (𝜑 → ⊥)) | |
4 | 3 | sb2imi 2080 | . 2 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]⊥)) |
5 | 2, 4 | mtoi 202 | 1 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ⊥wfal 1550 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 |
This theorem depends on definitions: df-bi 210 df-tru 1541 df-fal 1551 df-ex 1782 df-sb 2070 |
This theorem is referenced by: (None) |
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