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Metamath Proof Explorer
Theorem nsb
Description: Any substitution in an always false formula is false. (Contributed by Steven Nguyen, 3-May-2023)
|
|
Ref |
Expression |
|
Assertion |
nsb |
⊢ ( ∀ 𝑥 ¬ 𝜑 → ¬ [ 𝑡 / 𝑥 ] 𝜑 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
alnex |
⊢ ( ∀ 𝑥 ¬ 𝜑 ↔ ¬ ∃ 𝑥 𝜑 ) |
| 2 |
1
|
biimpi |
⊢ ( ∀ 𝑥 ¬ 𝜑 → ¬ ∃ 𝑥 𝜑 ) |
| 3 |
|
spsbe |
⊢ ( [ 𝑡 / 𝑥 ] 𝜑 → ∃ 𝑥 𝜑 ) |
| 4 |
2 3
|
nsyl |
⊢ ( ∀ 𝑥 ¬ 𝜑 → ¬ [ 𝑡 / 𝑥 ] 𝜑 ) |