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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 1781 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
| 2 | 1 | biimpi 216 | . 2 ⊢ (∀𝑥 ¬ 𝜑 → ¬ ∃𝑥𝜑) |
| 3 | spsbe 2082 | . 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 1538 ∃wex 1779 [wsb 2064 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 |
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-sb 2065 |
| This theorem is referenced by: sbn1 2107 noel 4338 sbn1ALT 36859 |
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