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Mirrors > Home > ILE Home > Th. List > hbn | GIF version |
Description: If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
hbn.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
hbn | ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbnt 1583 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) | |
2 | hbn.1 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 1, 2 | mpg 1380 | 1 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1282 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-5 1376 ax-gen 1378 ax-ie2 1423 ax-4 1440 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290 |
This theorem is referenced by: hbnae 1649 sbn 1867 euor 1967 euor2 1999 |
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