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Mirrors > Home > ILE Home > Th. List > hbnd | GIF version |
Description: Deduction form of bound-variable hypothesis builder hbn 1584. (Contributed by NM, 3-Jan-2002.) |
Ref | Expression |
---|---|
hbnd.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
hbnd.2 | ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
Ref | Expression |
---|---|
hbnd | ⊢ (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbnd.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | hbnd.2 | . . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | |
3 | 1, 2 | alrimih 1398 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓)) |
4 | hbnt 1583 | . 2 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜓)) | |
5 | 3, 4 | syl 14 | 1 ⊢ (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓)) |
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: (None) |
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