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Mirrors > Home > MPE Home > Th. List > hbn | Structured version Visualization version GIF version |
Description: If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 17-Dec-2017.) |
Ref | Expression |
---|---|
hbn.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
hbn | ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbnt 2129 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) | |
2 | hbn.1 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 1, 2 | mpg 1715 | 1 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-ex 1696 df-nf 1701 |
This theorem is referenced by: hbexOLD 2138 hbnae 2305 ac6s6 33150 hbnae-o 33231 vk15.4j 37755 vk15.4jVD 38172 |
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