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Mirrors > Home > MPE Home > Th. List > hbnd | Structured version Visualization version GIF version |
Description: Deduction form of bound-variable hypothesis builder hbn 2131. (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 1741 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓)) |
4 | hbnt 2129 | . 2 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜓)) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓)) |
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: (None) |
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