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Mirrors > Home > MPE Home > Th. List > nsb | Structured version Visualization version GIF version |
Description: Any substitution in an always false formula is false. (Contributed by Steven Nguyen, 3-May-2023.) |
Ref | Expression |
---|---|
nsb | ⊢ (∀𝑥 ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alnex 1783 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
2 | 1 | biimpi 215 | . 2 ⊢ (∀𝑥 ¬ 𝜑 → ¬ ∃𝑥𝜑) |
3 | spsbe 2085 | . 2 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑) | |
4 | 2, 3 | nsyl 140 | 1 ⊢ (∀𝑥 ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1539 ∃wex 1781 [wsb 2067 |
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 1913 ax-6 1971 |
This theorem depends on definitions: df-bi 206 df-ex 1782 df-sb 2068 |
This theorem is referenced by: sbn1 2105 noel 4329 sbn1ALT 35725 |
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