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Mirrors > Home > ILE Home > Th. List > hbnt | GIF version |
Description: Closed theorem version of bound-variable hypothesis builder hbn 1544. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
Ref | Expression |
---|---|
hbnt | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-4 1400 | . . . 4 ⊢ (∀𝑥𝜑 → 𝜑) | |
2 | 1 | con3i 562 | . . 3 ⊢ (¬ 𝜑 → ¬ ∀𝑥𝜑) |
3 | ax6b 1541 | . . 3 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
5 | con3 571 | . . 3 ⊢ ((𝜑 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝜑)) | |
6 | 5 | al2imi 1347 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑)) |
7 | 4, 6 | syl5 28 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1241 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-5 1336 ax-gen 1338 ax-ie2 1383 ax-4 1400 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 |
This theorem is referenced by: hbn 1544 hbnd 1545 nfnt 1546 |
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